Didn't watch the video, but what is mentioned in the article is correct - analytic continuation is used to extend this series to an analytic function on the complex plane, and it happens to be the Riemann zeta function. By uniqueness of analytic continuation (due to a theorem or lemma that when an analytic function is constant at an infinite number of points, it is constant everywhere), we know this is the only valid extension that makes sense to work with (consider being analytic similar to what it means for a function to be differentiable in the real numbers).