Q:

Why is having a NATO missile-defense launch system such a big deal for Russia?

After of its activation almost three years ago, from time to time, I keep hearing on the news about Russia making statements about the NATO missile-defense launch system placed in Deveselu (locally known as "Deveselu shield"), namely that it is a threat to Russia (source and source):

"Both the US and NATO have made it clear the system is not designed
  for or capable of undermining Russia's strategic deterrence
  capability," US assistant secretary of state Frank Rose told a news
  conference in Bucharest. "Russia has repeatedly raised concerns that
  the U.S. and NATO defense are directed against Russia and represent a
  threat to its strategic nuclear deterrent. Nothing could be further
  from the truth," he said.
The United States says it is in "full compliance" with the
  Intermediate-Range Nuclear Forces (INF) Treaty, the landmark Cold
  War-era nuclear accord.
The statement from the U.S. mission to NATO on February 8 came a day
  after Russia demanded the United States destroy a missile-defense
  launch system deployed in NATO member Romania in order to return to
  compliance with the INF.

The Romanian Foreign Ministry also had a reaction to the latest Russian statements:

(..) the ballistic missile shield at Deveselu is a strictly defensive
  military structure, explaining that his remarks come in relation to
  the Russian Defence Ministry having asked the US to destroy the Mk-41
  launching system in the composition of the Deveselu shield, on the
  grounds that it would violate the Intermediate-Range Nuclear Forces
  Treaty (INF).

From a layman's perspective it is quite hard to understand why a defense system would be a serious issue for such a large military power.
Question: Why is having a NATO missile-defense launch system such a big deal for Russia?

A:

Putin has repeatedly (personally) stated that his experts advise that defense-capable missiles can be re-purposed (he calls it "reprogram", obviously invoking the idea that the only change which would be necessary would be flushing of firmware of the missiles).  He, on more than one occasion, invoked the possibility that such "reprogramming" can happen without even the host countries, in Europe, noticing a change.
In the same breath, Putin also insists that any missile siloed too close to Russia can have its range increased in the future, as technology advances.  This claim is somewhat supported by historical precedent.  After the end of the Iran-Iraq war, Iran increased the range of the Soviet missiles (which it received from the USSR) to make them capable of striking Israel.  However, the trade off for increasing the missiles' range was a reduction in their precision. 
Putin has stated (in no uncertain terms) that the Russian position is that the US policy may change in the future and that any commitments made may be "reconsidered" (a clear euphemism for "reneged") by the US.  And that, as a result, any hardware allowed to be deployed in Europe may be upgraded (and that the US may upgrade it clandestinely).
NOTE
The links to Putin's speeches only make sense if you can get hold of someone who speaks Russian.  Sorry about that.  
As is clear from the 1st link, the "translation" is just a voice over restating the news program's narrative.  The actual words uttered by Putin in public tell a somewhat different story.  In the 1st link, he makes the claim that a change of firmware would not be noticed by Romanians.  I don't believe that this particular point was made in the voice-over "translation".  
I have heard him make the same claim about Poland when he was voicing his opposition to missile-defense interceptors in Poland, but I do not currently have a link to it.  

A:

Russians are worried that the GBIs (ground-based interceptors) can be nuclear-armed, and used for decapitation strike.
Jeffrey Lewis (of the Arms Control Wonk blog) had an article on one angle of this topic in Foreign Policy magazine in 2016, specifically about accusations about nuclear armed interceptors at Deveselu. In it, while dismissing most of this as Russian propaganda aimed at local Romanian population, he also cites his ealrier 2012 article which explains that Russians have a legit (to them, at least) concern:

It is a funny sort of paranoid fantasy, the notion that the United States might place nuclear weapons on missile defense interceptors and use them to decapitate the Russian leadership in Moscow. But I suspect this is the rub. The simplest explanation for Russia’s overwhelming concern with missile defense is that the General Staff fears that Russia is much, much more vulnerable to an attack against the country’s command-and-control infrastructure — what used to be called decapitation — than we realize. Part of this is a fear that missile defense interceptors could be armed as offensive missiles, part of it is that missile defenses could mop up a disorganized Russian retaliation. Most of it, however, is probably sheer terror at the persistent technological advantage held by the United States in light of Russian vulnerabilities.

Russia could be worried that the state of their strategic response is bad enough that high-efficacy missile-defense sites in Europe meaningfully affect their chances of response and therefore alter strategic balance and allow "reasonable" intentions about first strike without sufficient worry about counter-strike capability. That is a general opposition to almost any missile defense.
Obviously, there are objections to that line of thinking (see this RAND study, which also on top of its own conclusions cites Russian experts). But that doesn't necessarily convince top brass and politicians in Russia any more than technical research convinces their counterparts in the West.

A:

There are probably several reasons for the Russian government to be making these claims. Obvious possibilities include: 
They're talking to the Russian population, not the rest of the world
The Russian economy is not doing well, so it's worth focussing the population's attention on external threats. This will tend to distract them from discontent over their living conditions, the steady concentration of power and wealth in the government and its immediate allies, and so on. 
We don't know how healthy the Russian strategic deterrent is
The government is short of money, but they have a lot of missiles to maintain.
It's a negotiating strategy
If they make a fuss about a missile defence system, they might be able to get some concession in exchange for accepting it. 
The Russians don't regard the Iranian threat as credible
The missile defence systems installed in Poland a decade ago were explicitly for defence against Iranian missiles. The Russians didn't regard that threat as credible, mainly because they feel that the Iranian government is somewhat rational and capable of being deterred by US strategic weapons, and by French and UK SLBMs. 
So they don't see any threat that that the missile defence system could be used against, apart from Russia. 
They suspect the missile defence system can be rapidly strengthened
One claimed reason why the missile defence system "can't be intended for use against Russia" is that the numbers of missiles are too small to make a significant difference. However, rapid production of weaponry has been an American speciality in the past. The present-day procurement system is quite inefficient, but that could be changed. 

