Purgatory is one of those ideas that it can be hard to wrap one's head around, especially if one doesn't happen to be Catholic. After all, we are brought up with the idea of praying for the dead from the time we are young. Even if it isn't pointed out to us, "You should pray for the dead," almost every Mass we attend is offered for someone's soul. It is just part of the culture.
And why would we pray for the dead unless we could help them in some way? If they are in heaven, they certainly don't need our prayers. And if they are in hell, they are beyond the point where our prayers can be of service, so there needs to be a third option.
Still, if you happen to get into a conversation with someone from another faith, it is good to have some reasons for our belief in this place of purification. Meg Hunter-Kilmer offers a good explanation in this article: Purgatory. Prove It.