Cantor's argument also works in the finite case and this may serve to demonstrate the idea.
Consider 4-digit binary numbers like 1001. We can use Cantor's argument to show that there are more than four such binary numbers. Imagine you had a list of four such numbers, say 1001, 1010, 1110 and 0011. Then I can construct a number that can't possibly be in your list since it differs from the first number in the first digit, from the second in the second etc. In this case the number is 0100.