Tao's article certainly requires more background to read, but it gives a much deeper insight than explanations based on Cesaro or Euler/Ramanujan summation, which (as Tao points out in the article), in addition to being very puzzling, are actually mutually inconsistent. His main result is that these classical sums / the zeta function are only the
constant term in a certain asymptotic expansion associated with a smoothed version of the original sum. Recognizing the existence of the non-constant terms in that expansion corrects the nonsensical negativeness and logical inconsistency. This is intimately connected to the analytic continuation POV, which is discussed in section 2. It's well worth a read if you have time over a few days. I wrote an overview of the strategy here:
https://www.reddit.com/r/math/comments/cx3qzv/terence_tao_un...
