I don't think the three stars necessarily mean anything special. It's just a graphic design default. Stars represent states on the US flag, so whenever it's a patriotic type thing like the donkey/elephant for the two parties, they'll include stars just as a reference to that. If there was no context, and someone say a drawing of an elephant, they would just see an elephant. But an elephant with stars, well, that is associated with Republicans.

And one star by itself looks empty, odd number groupings tend to look better than even number groupings and 5 stars would be crowded, so three is a good visual balance. I'm sure there are a lot of people who would say it's supposed to reference the Holy Trinity or the three branches of our government (Executive, Judicial, Legislative) or something of that nature.

But honestly, I think it's just a basic graphic design choice to make it look more interesting.