Wait, what tactical or combat benefit is to be gained from culture?

It also spreads to other planets, affecting them.
It also keeps your planets from falling to enemy culture.
It also gains bonuses from some topics in your economy tech tree.
With enough of them, you can conquer enemy planets with pure culture. It tends to be a bad strategy, though.

IIRC it differs per race:
Advent: It gives extra Shield Mitigation
Vasari: It makes your weapons do more damage
Tec : It provides extra Antimatter regeneration

This is whenever you're in culture, whether in one of your gravity wells, a neutral one, or an enemy one. The amount of bonus it gives depends on the culture tech you have researched. (it should say which bonus on the research tooltip for the tech)

So, on one of the huge maps, there's desert planets near the empire boundaries. Playing as tec, I decided to put 10 broadcast towers on one.

20 minutes later, the four asteroid and volcanic planets nearby had turned coat...it was nifty

it can be a way to force an enemy who you have pushed back to 3 or so worlds in a big map to lose but it doesn't always work. If you get enough broadcast centers you can push through cap ships and start influencing population forcing them to deconstruct things like research centers to build media hubs. used it once in a game against a friend who used RA to hold out forever and it worked OK. Took a while but I had been beating my head against his fleet for 40 min already so I was ready to try something. It took a lot of hubs though and I had to jump in and kills a few cap ships to get it going. Also I was Advent so I used my super weapons to keep the broadcast centers he did have working for me. I never tried it again so I have no idea how it would work if it were used in other situations but it worked that night. he also didn't see it comming till too late.

Plus, just seeing the culture spread is just cool?