Citing from Wolfram Alpha:
"Any set which can be put in a one-to-one correspondence with the natural numbers (or integers) so that a prescription can be given for identifying its members one at a time is called a countably infinite (or denumerably infinite) set. Once one countable set S is given, any other set which can be put into a one-to-one correspondence with S is also countable."
The whole proof lies on the ground of one-to-one correspondence and the definition of equality based on it.
If one accepts that |N| = |Z| (the number of natural numbers equals the number of integers) then he/she should accept the proof shown in Numberphile too - as they do the same trick again and again.
Me personally feel some controversial feeling about it and it's an open question in my mind. But if the physicists have shown that the Riemann Zeta function at -3 IS EQUAL to zeta(-3) then it suggests to me that the Cantor method is working and we can say that if we find ANY one-to-one correspondence between 2 sets that they are equally large.
However it raises other questions like how should we imagine the numbers - whether a circle would be better where the infinity and minus infinity are basically the same point on the circle just right in the opposite side from zero? It's all just very weak imagination plays but there must be something...