Broadcast centers start spreading your culture from whichever planet they're built around, doesn't have to be the capital

Culture spread stacks with multiple centers around the same planet, but each jump (from one planet to another planet is 1 jump) decreases the amount of culture that keeps going. You don't need to build centers at each planet, but spreading some out around key planets will definitely help you spread it through your territory much much faster (after a few jumps, culture spread slows to a crawl).
As far as allegiance, regardless of how much culture you're spreading, you only get a 10% allegiance bonus for the affected planets, and allegiance gain/loss caps at +/- .05.
You might ask why bother stack lots of culture, then? And the simple answer is you don't have to unless you need to beat the enemy's culture and start dropping allegiance at his planets

Hopefully this helps!