This is a derogatory term used to express hatred towards split-screeners during online multi-player. As most COD gamers are aware, split-screeners are usually horrible and do more harm than good in any game type.
The meaning behind "Welfare Kids" is that the majority of split-screeners (some, not all) cannot afford their own PS3 or XBOX, so therefore in order to enjoy the COD experience, they play on their friend's console in split-screen. Its highly ironic though in some situations that the second player does better than the first player.
I, personally, have yet to play a round in TDM (etc) where a Welfare Kid has been on top at the end of a match. (One exception, college buddies of mine in FL were playing in the others dorm room and were split-screen playing, in which case they both did considerably well since they usually kick @$$ solo.)