According to Jim Recap: No Harm, No Fowl
When Andy is outside playing with the girls, he gets beaned in the head by a bird. Lucky for him he got hit in the head, where it shouldn't hurt him too badly. Gracie falls in love with "Daphne" immediately and wants to keep her forever. It's safe to say that no one else shares this opinion, and they rapidly try to find the owner by putting up fliers.
Unfortunately, the stick figure bird drawn by Jim nets only one potential owner (played by Jack Coleman, now on Heroes), but he quickly disavows owning the bird and that means that it looks like Jim is stuck with the bird.
He does have one other scheme up his sleeve, but it involves Kenny from work, and Kenny is not terribly bright, so it doesn't work out well, which means that it looks like they'll be keeping the bird - and when Jim sees Gracie's face when she finds out that he lied about Kenny being the bird's owner, he knows that he needs to try and make it up to her.
So when the bird escapes through an open window, he tries to do just that, by looking for it with her in the park, and when they find it, he even climbs a tree to try and get it back. That doesn't turn out well for him, but the bird does come home - and when they get there, the man who showed up earlier is back with his wife, who has been looking for her bird. Apparently he just didn't want it. Problem solved.
