Exposition is a term for the background information a story provides.
Wilde gives us the background of both the Happy Prince and the swallow. The Happy Prince is dead when the story opens and has become a grand statue in the town square. We learn that he lived a life of ease and comfort behind the walls of a palace. He thought only of himself and his own pleasure.
Now that he is a statue in the city, however, he sees all the misery and want he was shielded from in his palace. His heart cries out to help people, but he is immobilized because he is a statue. He may look happy, but inside he is sad.
The swallow stayed in the city after his fellow swallows left because he was in love with a reed, but she wasn't true to him. He decides to spend the night sleeping on the statue and then join the other swallows.
This explains how the statue and the swallow meet, and why the Happy Prince wants the swallow to help him care for the poor people of the city.