the bonuses come from research. each race has its own line of culture boosting research in its civil technology tree. each of these techs increases culture spread rate and also provides a combat bonus for ships in areas of friendly culture. 

advent culture tech - increases shield mitigation 

vasari culture tech - increases damage dealt

TEC culture tech - increases anti-matter regen

all cap ships, vasari ones just get an upgrade for it

That ability combined with the Deliverance Engine is a handy remote recoinnasance tool. One question, does friendly culture also give your ships your race specific bonus?

i think maybe ToJKa was referring to culture from allied players. i.e. does it stack?

the on-screen interface only shows 1 color of culture at a time going through a system so I always assumed you only get the bonuses that would be provided by that player's culture. however, it does seem that behind the scenes more than one friendly culture can be present in a system (if two allied players both have media hubs near the system) so its still an unresolved question. 

i'm leaning on no, only the highest culture (the one with its color displayed) applies bonuses. but i don't know for sure. 

From what I've seen in the past, your allies get nothing except increased alleigance.