Broadcast centers spread culture. Culture's primary effect is to change the allegiance rating of planets. If it's your culture and your planet, it will make the allegance go up to the maximum (which is 10% above what the planet starts at, 110% at the capital and drops from every planet hop away). Allegance controls how much income you lose on a planet, so boosting it is good.

Your culture on an enemy planet lowers allegance. Drop it to 0, and the planet rebels and the enemy loses control of it (but you do NOT gain control). High enough culture on a planet that's not yours will also block you from colonizing it until you can clear that out (Capital ships in orbit repel culture). So you can use it offensively by putting lots of broadcast centers on border worlds and trying to overwhelm enemy planets with your culture. Advent are the strongest at this due to more ways to boost their culture, but anyone can do it.

Culture's secondary effect depends on race. Each one has a line of technologies that give a secondary effect while in culture (TEC = Antimatter regen bonus, Advent = shield mitigation bonus, I forget the Vasari one). You get that while ships are in your culture, so it's to your benefit to fight in your culture and not enemy culture.

Culture will spread out multiple worlds past a broadcast center, but it decreases for each planet it passes through. That means if you're behind the line you only need one broadcast center to cover several planets. On border worlds if your enemy is pushing culture, you may need a lot more, or another way to repel it (Capital Ships, Advent Starbases with the culture module also repel and spread culture).

Far as I'm aware it's still possible to cause worlds to revolt with culture, as I've done it in Rebellion. You don't take them over automatically, but you never did. Nothing's really changed on that front.