How does this work?

This method is called the Luhn algorithm, here is a demonstration of how it works.

First, take all the digits of your credit card, and double every second digit from the right. If the digit is above 9, then you subtract 9 from it.

Digits of your card:

 

Every second digit from your card, doubled, starting from the right and subtracted by 9 if it's over 9:

 

Take the sum of every digit, even the unchanged ones:

 

If the number is divisable by 10, then it's valid, otherwise it is invalid.