Why does a swing come back down after going up?

A swing goes up one side, slows, and comes back again. It does not need a motor to begin that return. As the seat rises, gravity keeps pulling it downward.

The chains keep the seat on a curved path. From a high place on that path, gravity helps the seat move toward the bottom. It gets faster as it comes down. The chains guide the journey; gravity helps bring it lower.

At the bottom, the seat is still moving. That movement carries it up the other side, where it slows again. The seat does not stop just because it has reached its lowest place.

An empty swing eventually makes smaller trips. Air pushes against its motion, and parts rub where it hangs. Without more energy being added, it settles near the bottom. The back-and-forth journey is a repeated exchange between moving fast and being high above the ground.