It's not that the moon's mass changes (it doesn't) but the positioning of the moon in space-time relative to the sun and earth that minutely changes its gravitational field/influence
. Gravitational forces are functions of both mass and
location (we weigh less at high altitude than we do at sea-level because we are further from the focus of the Earth's gravitational force). Picture the Universe as a suspended piece of fabric pulled tight. Drop a ball (the sun, for instance) in the center and it causes a funnel-like depression in the cloth. You now drop other smaller marbles onto the cloth and they start spinning around the sun's divot (like one of those cool little games where you drop the quarter into the chute and it spins around the funnel before dropping into the hole in the center). These marbles will mainly follow the depression (gravitational pull) of the sun but will also form their own little divots as they travel. Though these divots are relatively small, they still interact in the motion of other nearby marbles (planets, etc.).
Sorry, just passing the time before I leave work to go boating.