If you just dumped the sugar straight into your bottling bucket, it's not going to dissolve very well or mix very thoroughly into the beer, so you'll wind up with unevenly carbonated bottles. So, the sugar has to be dissolved into something that will easily mix in with the beer.
Now, sugar doesn't dissolve very well into cold water. Also, water (especially tap water) tends to have a lot of air in it. Boiling the water makes the sugar dissolve easily into it AND it also drives out that air and oxygen, so you won't risk oxidizing your beer when you introduce the priming sugar.
Also there's that whole sanitizing thing, but that's not 100% necessary as you can tell since you don't really sanitize the sugar you use when bottle priming.