The paradox described in the play is that Frederic is both 21 and not 21 because he was born on a leap year so he will not actually have a 21st birthday, even though he’s been alive for 21 years.
A paradox is something that is true and not true at the same time, or something that is a contradiction.
Frederic is a young man who has been with a group of pirates for years. His contract states that he will be done with his contract when he is 21, on his 21st birthday. He was supposed to be contracted to a pilot, but alas his nurse accidentally heard wrong and accidentally contracted him to a pirate. Whoops. The pirate gang is pretty bad a being pirates, because they let anyone go who claims to be an orphan. Word gets around, and everyone they meet says he’s an orphan. So Frederic has a pretty easy go of it. On his 21st birthday he is pleased to leave nonetheless, especially when he meets the girl of his dreams. Unfortunately, Ruth and the pirate king burst his bubble by telling him the paradox. He was born on a leap year, so he will not have a 21st birthday even though he has been alive for 21 years. He is both 5 and 21 at the same time.
How quaint the ways of Paradox! At common sense she gaily mocks! Though counting in the usual way, Years twenty-one I’ve been alive, Yet, reckoning by my natal day, I am a little boy of five! (Act II)
A leap year means that you only have a real birthday every 4 years. So Frederick has not really been having birthdays these 21 times. He has only really had 5. Yikes! His sense of honor and duty tells him he has to stay until he has had all 21 leap year birthdays.
FRED In 1940 I of age shall be, I’ll then return, and claim you – I declare it!
MABEL. It seems so long!
FRED. Swear that, till then, you will be true to me. (Act II)
Well, I guess true love will wait for anything! It is a silly device of course, but the play is not intended to be serious. Frederic's honor is at stake, not the letter of the law. He stays not because he has to, but because he feels he needs to in order to stick to his word. Eventually, the pirates and the girls form a kind of truce on the convenient device that all of the pirates are just noblemen who have gone astray, and everyone gets married and lives happily every after!