While in freeburn (multiplayer free-roam), if you Takedown another player's car or or receive a Takedown from another player, it will spark a 'rivalry' between you. It will also get you a number of mugshots toward the Hotshots in Burnout Paradise
achievement if the players you are taking down have a camera.
A Takedown simply requires you to total someone's car, by hitting it with enough force to total it upon impact or by forcing it into a wall. Grinding it along a barrier/wall for long enough or landing on it from a height also count as Takedowns.
I have seen it mentioned in other guides for some reason, but you should not require a vision camera for this at all.
Keep playing and you will get this one before long, assuming you don't get it during normal play.