These are some quick sketchnotes based on this lecture by Ryan O’Donnell, a part of the playlist for the awesome CS Theory Toolkit course. You can walk through the arguments below.
I should mention that while the Schwartz-Zippel-DeMillo-Lipton lemma is invoked in the notes below, one could make do with just the fact that over any field , any degree polynomial has at most roots, as pointed out by @dsivakumar — thanks!