How does the moon stabilize earth's axial tilt?
The stabilization comes from the tidal interaction between the Earth and the Moon.
The same tidal interaction that makes high and low tides line up closely with the Moon, does the same thing with the bulge at our Equator; in effect the bulge is like a big high tide that tends to favor facing the biggest tidal force -- which is from the Moon. The Equator is pulled close to the plane of the Moon's orbit, and the poles follow suit.
For a more complete description of how the Moon and the tides induced by it influence the Earth, see here: