> Could someone explain what the IAU means by "clearing its orbit".

To use a different phrase from wikipedia, "orbital dominance".

Other objects that stick around are either captured as satellites or forced into resonant orbits.

> The Sun attracts the Moon more strongly that Earth does, so does that mean that the Earth doesn't fit the criteria?

The Earth's sphere of influence goes out to about a million km, 2.5x the moon's orbit.