How exactly does culture effect planets?
It raises the allegiance cap of affected planets by 10% and increases the rate at which your allegiance rises to to match that cap. Higher allegiance nets you a higher percentage of the resources a planet and surrounding metal/crystal deposits can possibly offer (100% allegiance = 100% of possible resource income).
What would the benifit of immersing an enemy planet in your culture?
If an enemy planet comes under the influence of your culture, they'll lose their own culture bonus and start losing allegiance until it hits 0 at which point, the planet is abandoned. Even if you don't take out the planet this way, the allegiance hit will cut into your opponent's resource production.
In most games, this isn't a very effective way to destroy planets as it takes a 'lot' longer than simply bombing the planet as you noted. In cases where the planet is heavily fortified or when you're using the Deliverance Engine (the Advent superweapon), it's still one backdoor you can take which will force your opponent to respond in some way.
A couple other notes on allegiance and culture:
- Allegiance is governed by distance (in phase jumps) from your current home planet. If you want higher allegiance throughout your empire, set your home planet to a more central location. The only other way beyond this and culture to raise allegiance is with a certain Tier 8 research option with the Advent faction.
- Culture offers several race-specific bonuses when you research the culture-boosting upgrades. The TEC get a bonus to anti-matter regeneration, the Advent get higher shield mitigation, and the Vasari get a damage bonus. Getting these buffs and negating them on your enemies is another benefit of spreading culture.
- Capital ships naturally slow the spread of enemy culture simply by being in the path of the spread.
- The Advent, with appropriate research, are able to reveal zones that are under the influence of your culture which is an excellent supplement to scouting (especially if you have Deliverance Engines on the field).