Why did they fight? Kronos ate all of Zeus' brothers (not the sisters as far as I know, as the prophecy was specifically directed at Kronos' sons.)

Kronos only did so because of a prophecy that said that he would be defeated by his sons, as he himself defeated his father. So he removed his sons from existence. Personally, I would've thought some form of protection/abstinence would have worked better, but hey, what can you do.

So basically it was revenge motivated, plus a bit of power struggle in there as well. The Greeks didn't put too much into why I think.

If you wanna go all analytical on the myth, then it's a representation of how sons are destined to fight with their fathers for dominance (even if not physical blows, just arguing), and how, eventually, the youth will overthrow the elders. And then the cycle will continue, always- the sons become fathers, and their sons will fight them.