Q:

Why has Donald Trump's approval rating been rising since impeachment processes started?

It appears that recently, Trump's approval rating has been rising despite  impeachment, and the accusations and evidence against him. 
How can this be explained? Has there been anything published about why the public continue to support him, and indeed seem to be becoming more likely to do so?

A:

From the very same article you cited about the very same poll that showed that Trump’s approval is up by about 6 percentage points:

Support for impeaching and removing the president has dipped slightly among independents, Gallup found, from 55 percent and 53 percent in two October polls to 48 percent in the latest survey.

If you split the difference between 55 and 53, you get 54, which is 6 points greater than 48.
Given that the drop in support for impeachment is exactly proportional to the increase in support for the President, and that there’s very little else in political news being reported at the moment, the simplest explanation is that independent voters are not being persuaded by the impeachment case that the Democrats in Congress have put forward.

A:

Impeachment hasn't substantially improved Trump's approval rating.
Approval rating fluctuates for every president, as it's based on (a tiny subset of) the people's opinion, so you have to get the average of not only multiple polls, but also across multiple measurements in time, to see if it's actually trending up and trending down.
This is the historic trend of Trump's approval rating (black line) and disapproval (red line), over the past two years.

Here's me roughly averaging it his approval polling, by eyeballing it (blue line).

Here's me adding in the start and end of the Russian investigations, and the start of the Ukraine scandal and impeachment:

When you look at the broad picture, you notice that as he got ensnared in the Russian investigation and continued criticism and bad press over treatment of illegal immigrants, he dropped in approval, but it has since balanced out and has mostly held steady over the past year-and-a-half, with a small gradual upward trend as the economy has continued to do well. By "holding steady", it still has the regular up and down bouncing of people's opinions in the average of a dozen tiny polls measuring a thousand people here, a thousand people there, out of a nation of >325 million.
Trump's gradual upward trend pre-dates the impeachment process, by over a year.
Further, the fluctuations are of a few tiny points here or there, e.g. 1 out of a hundred people changing their mind slightly from "Extreme disapproval" to "Mild disapproval" or from "Neutral" to "Mild approval". These are not big swings of the population, if you look at the scale on the left side of the chart.
The impeachment hearings were rushed through so quickly, that it's really too short of a polling period to filter out the regular fluctuations of Trump's approval from the effect of the impeachment.
The impeachment hasn't had a substantially noticeable impact one way or another, with the exception of a small blip right when it began.
In my interpretation of this data, I'd say that it's not that the impeachment has increased Trump's approval, it's rather that the impeachment hasn't substantially decreased Trump's already very-very-very slowly increasing approval trend that started over a year prior to impeachment.
That much I can discern by looking at the raw polling data. So, the question becomes not, "Why has impeachment helped Trump?" (which it doesn't seem to have - at least, not enough evidence exists to substantiate that), but rather "Why hasn't the impeachment hurt Trump?".
To answer that new question, we have to enter speculation, which obviously gets distorted by our personal biases.
Why hasn't the impeachment hasn't decreased Trump's approval? [Speculation / Personal opinion]
Any change in the polling we're measuring are mostly people who are neither concretely locked into being pro-Trump (concreted pro-Trumpers are solidly 38% of the country who've supported him without fluctuating) nor people concretely anti-Trump (solidly 42% of the country who haven't wavered against opposing him).
I would speculate, based on my biased interpretations of the trends visible in the charts above, that voters that are neither permanently Pro-Trump nor permanently Anti-Trump were really bothered by, and concerned about, Russian interference, and that the Russian investigation hurt Trump with these voters. The national narrative was, "Trump himself explicitly colluded with the Russians to win the election, and is working to impede the investigation."
But after the Russian investigation ended there was little concrete evidence that Trump colluded with the Russians (though he did impede the investigation in my opinion, but Mueller refused to issue a "verdict" on that), and non-concreted Americans mostly thought Trump was vindicated on the primary charge: outright collusion with Russia.
Then, when the Ukraine scandal began just a month or two later after the Russian investigation ""cleared"" Trump (or at least didn't convict him badly enough to be impeached), to these independents, it possibly failed to work against Trump, for several reasons:

First, to some, it looked like Democrats failed to convict him on outright
collusion with Russia, and now were just throwing something else at
him to try to get him out of office.
Second, what they were now impeaching him over ("clumsily abusing his office 
to get an edge over Biden in re-election") pales dramatically in
comparison to "treason against the United States by colluding with our
greatest enemy".
Third, Joe Biden has terrible optics here. Biden is on video openly bragging about removing a Ukraine investigator investigating a Ukraine company that his son, with zero relevant experience, was getting paid $600,000 a year by. (This was not Biden's motivation, and is missing nuance that vindicates Biden, but that's the narrative many were hearing). It doesn't help that Biden publicly stated he never discussed his son's Ukraine work, followed by Trump's campaign releasing photos of Joe Biden with his son and his son's Ukrainian boss golfing together, and these photos went viral. Trump putting
pressure on Ukraine to reopen that investigation can, by squinting in
the right light, be seen as almost reasonable to independent voters.
Independents here aren't sympathetic to Biden, as it looks very "swamp-like" on the surface.
Fourth, the Russian investigation went on for months and months and
months, and seemed really official and thorough, and was conducted 
independently by FBI, at the request of congress. In contrast, the Ukraine 
impeachment sped by in just two months, entirely controlled by a single 
political party, and the voting was entirely along partisan lines. No matter 
how neutral, fair, or unbiased they may have been, the optics looked partisan 
as hell in comparison to the Russian investigation.
Fifth, Trump's phone call that was the focus of the impeachment is
rather ambiguously worded. In typical Trump fashion, he rambled around non-stop, jumping between multiple topics. To impassive third parties, it just sounds like clumsy Trump speaking clumsily. The reasoning goes, Trump is like a bull in a china shop, and if he was trying to do quid-pro-quo, he likely wouldn't have been subtle.
Sixth, the Prime Minister of Ukraine, who the articles of impeachment
claims was being "pressured" by Trump in a quid-pro-quo, openly
claims he wasn't being pressured, and it wasn't a quid-pro-quo. Sure, he could just be not wanting to rock the diplomatic boat, or even trying to win favor with Trump, but him publicly saying that does harm the perception of the impeachment process to undecided voters.
Seventh, several expert Democrat witnesses the Republicans had provided to testify during the impeachment process - professors of law at various universities - very smoothly and publicly articulated why they opposed the impeachment, and why they, as Democrats, thought the Democrats were in the wrong. Accurate or not, that harms the optics of the impeachment process.

(Note: I keep using the word "optics", because we're talking about why public polling of random citizens many shift one way or another. In such cases, optics is everything, as unsubstantiated opinion is exactly what the polls are measuring).

Eighth, it's come out recently that the CIA made several huge mistakes during the past Russian investigation, that cast the Russian investigation in a bad light, and make the Russian investigation look more partisan in hind-sight. These revelations occurred during the impeachment process, and their effect on the polled public's perceptions can't be separated from the effect of the impeachment hearings themselves. For example, James Comey himself has been publicly admitting he botched part of it, when getting the FISA warrants to even begin the investigation.
And finally, a few Democrat congressmen have openly said they need to impeach Trump so he doesn't win re-election. Giving the impression that the impeachment is just a partisan attempt to derail Trump, in the same way they are claiming Trump was abusing powers of office to derail Biden. Sure, hundreds of Democrats may have been taking the impeachment process in a perfectly unbiased and neutral way, but if even one or two Democrats publicly say they themselves are being partisan, it gives the impression all the two hundred others silently are also.

All this adds up so that people who are neither rabid anti-Trump nor rapid pro-Trump could feel far less worried about the Ukraine scandal than they did about the Russian scandal, and isn't letting it tamper their slightly rising approval over the good economy.
Note: As the question itself is one that can only be answered by speculating individual voters opinions, I've tried to provide reasons why they might have leaned more Trump-ward over the impeachment process.
No concrete answer can be given without examining the unique individual minds of the voters polled.

A:

The easiest explanation is that it's a matter of who people believe. The Republican stance on the impeachment has been that it is a partisan hitjob intended to oust a democratically elected President for no justifiable reason. They're using Pelosi's own words against her from when she stated that she did not want to move forward on impeachment unless there was strong bipartisan support. And unfortunately, there has been zero bipartisan support, and there has even been dissent within the ranks of the US House of Representatives, with one Democrat expected to switch parties in January. He and one other Democrat from a strongly red district voted against impeachment.
The other defense talking point has been to read the transcript of the Ukraine call. These next bits are primarily conjecture and personal opinion, with limited documented support, so please take them with a grain of salt; The transcript is too long and dull of a read for the average voter. It's 5 pages of largely dull conversation that, without a decent knowledge of external factors (the way Congress had already approved aid, the other conversations denying a white house visit, etc.) doesn't look all that nefarious. Most people don't really want to sift through the document-- they want a snappy excerpt from it that supports the stance they've already taken, and they're inclined to believe whoever they hear that has agreed with them in the past. As such, their views on impeachment are those of whichever broadcasting network they've watched the most.
The final factor (and the one that I believe influences impeachment views the strongest) is that Senator Mitch McConnell and Senate Republicans have made it clear that they intend to make the impeachment trial as swift as possible, and to settle on no convictions ASAP. The Republicans have a simple majority in the Senate at this time, and that's all it takes to override the objections of the minority party and deny admittance of evidence that they do not wish to see. Many believe that this means McConnell will block key witnesses who would be exceptionally damaging to the President, including those who have been barred from testifying in the House's prior investigations. When Bill Clinton was previously impeached, his support grew following the Senate's failure to convict him. Many persons who do not support Trump oppose impeachment because they believe that a failure to convict will strengthen his run in 2020 and could lead to 4 more years of Trump in the White House.

