Open the Battle and go to the Games tab. You'll see the games and markets included in that Battle's Battle Slate. Pick a market, enter an amount from your Battle Bankroll, and submit. It works just like making Picks in a Challenge.
Once a Pick shows up in your Picks tab, it's final and can't be canceled or changed, so double-check before you submit.
If a Pick ends in a tie (a push), the amount you put on it goes back to your Battle Bankroll. For games that are postponed or rescheduled, see What happens to my pick if a game is postponed or rescheduled?