I'm no expert but I think the trick is to say "give me a function that for any element in Set A, returns a matching element in Set B".

For mapping integers to even numbers it's f(x)->x*2

But for mapping integers to real numbers - there is no possible mapping.

Cantor's Diagonal Argument is really easy to follow - watch a YouTube vid or two on the topic. Also Hilbert's Hotel is well covered.