A re-entry permit is like a special passport for people who are not United States citizens, but who need to leave the United States temporarily and then come back later. It is a way for these people to show that they are still allowed to come back into the United States after they leave.
Think of it like this: if you go to a friend's house to play, you might need to tell your mom or dad where you are going so they know where you are. But if you are planning to leave your friend's house to go get lunch with your family and then come back later, you want to make sure your friend's parents know that it's okay for you to come back. That way, when you come back later, they will recognize you and let you in.
That's kind of how a re-entry permit works. If you are not a United States citizen, but you are here visiting, studying, or working for a little while, you might need to leave the United States for a short time and then come back. A re-entry permit is a document you can apply for that shows immigration officials that you are allowed to come back into the United States after you leave.
So it's kind of like a permission slip for grown-ups, to make sure they can come back to the United States later if they need to leave for a little while.