Hey, I have a better explanation after rewatching the video.
In the video he considers three sums
S1 = 1 - 1 + 1 - 1 + ...
S2 = 1 - 2 + 3 - 4 + ...
S = 1 + 2 + 3 + 4 + ...
Given such a sum, we can write a "generating function". That is, a power series using the entries in the sum as coefficients.
f(x) = 1 - x + x2 - x3 + ...
g(x) = 1 - 2x + 3x2 - 4x3 + ...
h(x) = 1 + 2x + 3x2 + 4x3 + ...
Notice that, naively speaking, f(1) = S1, g(1) = S2, and h(1) = S. These aren't quite true, since these three power series each diverge at x=1.
f(x) and g(x) each converge for |x| < 1. Here, f(x) converges to F(x) = 1/(1+x). Since g(x) is the term by term derivative of -f(x), we have that it converges to G(x) = -F'(x) = 1/(1+x)2. Note that F(x) and G(x) are analytic continuations of f(x) and g(x). That is, they agree with f(x) and g(x) whenever f(x) and g(x) converge, but F(x) and G(x) are also well-defined at some other values. In particular, F(1) = 1/2 and G(1) = 1/4. That is the sense in which S1 = 1/2 and S2 = 1/4.
By the way, here I used a derivative to compute G(x). In the video, they instead look at S2 added to itself with a shift. In terms of power series, the shifted S2 is really 0 + 1 - 2 + 3 - 4 which has generating function xg(x) rather than g(x). So the video's observation is really that g(x) + x g(x) = f(x). Thus G(x) + x G(x) = F(x), so (1+x)G(x) = 1/(1+x) and G(x) = 1/(1+x)2. No derivative necessary.
Now the next trick they consider is S - S2. Using our power series, the observation is that
h(x) - g(x) = 4x + 8x3 + 12 x5 + ... = 4x(1 + 2x2 + 3x4 + ...) = 4xh(x2).
Now if h(x) has an analytic continuation H(x), we have
H(x) - G(x) = 4xH(x2)
Plug in x=1 and you get
H(1) - G(1) = 4H(1)
Solving for H(1), you get H(1) = (-1/3)G(1) = (-1/3)(1/4) = -1/12.
