asteroids - What is the connection between moons, rings, and gaps?

It's mainly by the clear definition of the rings, and their mere existence.

Without a moon the rings would be short-lived, hence unlikely to be detected just in time shortly after they've formed. And they would tend to wash out to broader rings. The space between the rings seems to be empty. That's easier to explain by one or more moons keeping the gap clean, and keeping the rings on their track (shepherd moons).

A way of thinking of the shepherd moons:
Our Earth's moon slows down Earth's rotation by tidal forces and friction.
Think of a ring a little inside the orbit of a small moon as an analogon of Earth inside the orbit of Moon: It's slowed down by tidal forces and friction (non-elastic collisions between ring particles). Slowing down ring particles results in an orbit closer to the planet/asteroid. This way a small moon can act repulsive to a nearby ring, confining it.

More details about the dynamics of planetary rings, including confinement by shepherd moons, see this paper, p. 491 ff.