It doesn't just prove that there's infinitely many irrationals -- that is trivially easy to prove. For instance we know pi is irrational, therefore 2pi, 3pi, 4pi, 5pi,... etc are all irrational as well.

Cantor's proof proves that there's no one-to-one correspondence between reals and integers (whereas there is a one-to-one correspondence between integers and rationals), and therefore the infinite set of reals is "bigger" than the infinite set of rationals.

That there is an infinite number of irrationals should follow as a trivial consequence of this. (Left as an exercise for the reader...)