When you are studying a Deck in Brainscape, all you have to do is rate your confidence on the 1-5 scale (on the Answer side of the flashcard), and it should automatically advance to the next flashcard. The system automatically determines which flashcard should come next, based on your previous confidence ratings in the other cards in the Deck.
If you simply want to be able to swipe to the next flashcard on your iPhone, then you can consider using the Browse feature.
[See also: How should I rate my confidences? (When should I rate a 5?)]