What mistakes, if any, were made in Numberphile's proof that $1+2+3+\cdots=-1/12$? This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in Numberphile's proof (reproduced below). (Numberphile is a YouTube channel devoted to pop-math and this particular video has garnered over 3m views.) 
By general audience, I mean the same sort of audience as the millions who watch Numberphile. (Which would mean, ideally, making little or no mention of things that a general audience will never have heard of - e.g. Riemann zeta functions, analytic continuations, Casimir forces; and avoiding tactics like appealing to the fact that physicists and other clever people use it in string theory, so therefore it must be correct.)
Numberphile's Proof.
The proof proceeds by evaluating each of the following:
$S_1 = 1 - 1 + 1 - 1 + 1 - 1 + \ldots$ 
$S_2 = 1 - 2 + 3 - 4 + \ldots $ 
$S = 1 + 2 + 3 + 4 + \ldots $ 
"Now the first one is really easy to evaluate ... You stop this at any point. If you stop it at an odd point, you're going to get the answer $1$. If you stop it at an even point, you get the answer $0$. Clearly, that's obvious, right? ... So what number are we going to attach to this infinite sum? Do we stop at an odd or an even point? We don't know, so we take the average of the two. So the answer's a half." 
Next:
$S_2 \ \ = 1 - 2 + 3 - 4 + \cdots$
$S_2 \ \ = \ \ \ \ \ \ \ 1 - 2 + 3 - 4 + \cdots$
Adding the above two lines, we get:
$2S_2 = 1 - 1 + 1 - 1 + \cdots$
Therefore, $2S_2=S_1=\frac{1}{2}$ and so $S_2=\frac{1}{4}$.
Finally, take
\begin{align}
S - S_2 & = 1 + 2 + 3 + 4 + \cdots 
         \\ & - (1 - 2 + 3 - 4 + \cdots)
         \\ & =   0 + 4 + 0 + 8 + \cdots 
         \\ & = 4 + 8 + 12 + \cdots 
         \\ & = 4S
\end{align}
Hence $-S_2=3S$ or $-\frac{1}{4}=3S$.
And so $S=-\frac{1}{12}$. $\blacksquare$

This is a very good question. Most responses to the video instruct the viewer to use zeta function regularization or Ramanujan summation, without actually engaging the proof and showing why it's wrong. Or their demonstrations why it's wrong ignore the entire body of summability methods for divergent series.
Although they are not careful in the video to discuss notions of convergence, and which steps are compatible with which classes of divergent series, some of their steps can be justified. For example, the series $1-1+1-1+\dotsb$ is Cesàro summable, which is a stable and linear summation method, which means that the computation $1-1+1-1+\dotsb = 1 - (1-1+1-1+\dotsb)$ is justified, which shows that $1-1+1-1+\dotsb = 1/2$ (that computation isn't in the Numberphile video you linked, where they just wave their hands and say "average it", but there is also an embedded link to another Numberphile video where they do show this computation).
Similarly the computation showing $1-2+3-4+\dotsb=1/4$ can be justified by appealing to Cesàro summability.
However the series $1+2+3+\dotsb$ is not Cesàro summable, nor indeed can any stable and linear summation method sum this series without giving rise to inconsistencies. See for example this wikipedia heading or this example computation by robjohn. So when they do the subtraction $(1+2+3+\dotsb) - (1-2+3-4+\dotsb) = 0 + 4 + 0 + 8 + \dotsb,$ they are invoking linearity; they assuming a linear summation method. Then when they write $0 + 4 + 0 + 8 + 0 + \dotsb = 4 + 8 + 12 + \dotsb$ they are assuming stability. This is the error. It is inconsistent to assume both properties.
Edit: As pointed out by Mathologer, in fact stability allows you to insert only finitely many zeros into a series. So this would be an error even in the presence of Cesàro summable series.
I wonder whether their proof can be repaired by dropping the use of one of the two axioms, linearity or stability. I couldn't do it, so I asked here.

Edit: Today (13 Jan 2018) Mathologer posted a new video which I liked very much. I hope it doesn't sound like tooting my own horn if I say it seems to be a lengthy recapitulation and explication of exactly all the points in my answer here. He does improve on my answer in one way: he points out that to insert infinitely many zeros into a sum, as is done in the Numberphile video, is not valid even with a stable summation method. I called out that step as a mistake, but only because it was invoked on a sum not known to admit a stable summation method. In fact it would be a mistake anyway.
