What's the point of fences in Harvest Moon?

According to this FAQ, fences simply look nice. If you like how a farm looks with a fence, keep 'em around and maintain them. If you don't care, don't use 'em!

I always used them to separate parts of my farm for organizational purposes.

From a GameFAQ's article:

When it rains or snows, there is a chance that some pieces of the fence will break. You should always thoroughly inspect the day after a storm. Broken pieces can be removed with the hammer and then replaced. If you have any broken pieces, there is a chance wild dogs will come at night. The dogs are characterized by high pitched barking after you go to bed, and they will eat any chickens you leave outside (you will find chicken feathers the next day).
The more broken pieces, the greater the chances that dogs will come. Once you exceed about 15 or so broken pieces, the dogs will come every night until the fence is fixed.

I can verify that I have experienced this in-game.