Capital ships suppress culture. If you clear out the planet and wait around for a while it will clear out. If you don't want to wait that long:
1. Advent Starbases can be given the ability to both spread and suppress culture. Build one at the planet (Starbases use a special constructor ship and you can build them even if you don't own the planet) and buy the culture upgrades. That'll also help you if you want to conquor the neighboring systems by giving you some local culture generation. I don't remember if the other two factions can do the same thing.
2. The Advent Superweapon can spread culture on any hostile planet, fire that off on the neighbors.
3. If the neighboring planets have their own culture generation in mass quantities, you may simply need to go around and take out one or two more before you can colonize anything. Once you have a beachhead in the area you can bulid your own culture and it gets a lot faster.
