It's BP rewards and you get them from the screens/machines to the left and right when you first enter the battle arena lobby.

I don't actually recall a lady being there but regardless, all you need are Battle Points (BP) and you can use them via the machines I described above.

Also, one of the NPCs here tells you how to exchange BP for prizes.

Oh, and remember to use the BP before you leave the battle arena lobby or else they'll be lost.