When she gets excited she starts talking fast, which made it a little hard for me to follow her, especially when she gets to the nitty-gritty about how the square root of two is irrational, so I transcribed that bit, starting at the 5:38 mark (P is the Pythagoras voice and V is Vi playing the spoiler):
V: Just that you know an odd number like seven isn't going to be it without even trying. An odd number times itself gives an odd number of squares, so whatever this number is it can't be seven, it has to be even.So there are numbers besides whole numbers and rational numbers, there are also irrational numbers, which says something about our worldview, but I'm not quite sure what.
P: OK, so the hypotenuse is even. That's fine.
V: So what if I prove the leg is even too?
P: Then it's not in simplest form. Any ratio where both are even, you divide by two until you can't divide any more because one of them is odd and then that ratio is the best. I thought we assumed we were talking about the simplest form ratio.
V: We are. If there is a ratio in simplest form at least one of the numbers is odd and since the hypotenuse has to literally be divisible by two then the leg must be the odd one. So what if I proved the leg had to be even?
P: You just proved it's not. It can't be both.
V: Unless it doesn't exist! What you forget, Pythagoras, is that if this is the square then the two sides are the same. Just as this is divisible right down the center, so too is it divisible the other way. And the number of squares on this side, which are the number of squares in just one leg is an even number and for a number of squares to be even what does the number have to be, Pythagoras, oh my brother?
P: If leg squared is even then the leg is even, but it can't be even because it's already odd.
V: Unless it doesn't exist!
P.S. Hippasus is the name of the spoiler who Pythagoras might have killed.