Although the diagonal theory is an excellent one some people may find it conceptually difficult, so here is a way of showing that there are infinitely more real numbers than rational numbers.
Suppose you made a number line that was 1 foot wide and contained all numbers; it would be infinitely long and therefore it would have infinite area.
Now suppose you took a list of all the fractions (for how to list them see "infinity", and now suppose you took 1 square foot of paper. Cut it in half and put one half over the first fraction in the list, namely 0/1 or just 0. Now you could take the remaining paper, cut it in half, and put one half over the next fraction; 1/1.
You could continue this process until you had covered all the fractions in the list, although you would soon be adding strips of paper less than a billionth of an inch wide. They would still be wide enough because on a number line no rational number has any width at all.
This shows that while you need an infinite amount of paper to cover all real numbers on a number line you can use a finite amount of paper to cover all fractions.
If you need infinitely more paper to cover all real numbers than you do to cover all rational numbers then there must be infinitely more real numbers than natural numbers.
This doesn't constitute a mathematical proof but I found it made it easier to believe the diagonal argument.