Aang is climbing to the top of a mountain, but along the way, he must avoid Fire Nationsoldiers who will try to stop him at all costs.
The goal of the game beyond this is unclear; it is impossible for Aang to reach the top of the mountain. The most logical assumption to be made judging by the way in which points are earned (the more feet hiked, the more points gained) is that guiding the Avatar as far up the mountain as possible is the ultimate objective.
Aang will need to avoid firebenders, falling rocks, and trees, as there is a penalty to be paid when he is touched by them. If he defeats three Fire Nation soldiers in a row, Aang will glide on his staff and collect points at a faster rate. Sometimes a circular, hollow bubble will appear, and touching it enables Aang to act within a protective shield until he is hit by an obstacle, at which point he returns to normal.
To "take out" enemy soldiers, Aang simply needs to jump at them from the other side of the chasm.