The song is about going sleighing. The sleighs are pulled by horses wearing sleigh bells. (Actually, by one horse. That's what makes it a 'one horse, open sleigh.) It was the fashion at the time to 'bob' the tails of the horses - that is, cut them short. The line actually reads "Bells on bob tails ring'. Thankfully, people realized that bobbing tails was cruel, because horses need their tails to swish flies, and they stopped doing it.
So, both the previous answers are completely wrong. And "making spirits bright" just means that everyone feels uplifted (their spirit is made bright) by hearing sleigh bells.
Also, sleigh bells weren't worn on the tail. They're on a thick strap that I believe was strapped onto the harness.