Take any 13 people. There are only 12 birthday months. However you sort those 13 people into 12 month-boxes, at least one box gets two people. No trick, no luck. With more items than boxes, a repeat is forced.

This is the pigeonhole principle: put n items into m boxes where n is bigger than m, and at least one box holds more than one item. It sounds almost too simple to matter, but it proves things instantly that would otherwise take real work to show.