Natural bridges are formed by water eroding a new way though the rock.

This picture is from the NPS nature and science page

The river forms an “S” as seen in the first step. This happens when a disturbance (Muskrat and beaver homes, lava flows, landslide) changes the water flow.  The water ricochets from bank to bank creating an “s” shape.

Here is a great Minute Earth video about this.

The river cuts deeper into the earth making a canyon.  The ‘s’ becomes more pronounced. The curves come closer to each other, separated by a thin wall of rock.  The river erodes a hole in the wall and then flows through it forming a natural bridge.  The old riverbed can then become an oxbow lake.  Over time the natural bridge collapses.

In Natural Bridges National Monument, bridges will form and collapse over time.