Why does the full moon always rise at sunset?

Think of it this way. In order for the Moon to appear fully illuminated by the Sun you have to be in between them. In other Words when the Moon is full, the Sun is opposite it in the sky, and we are in the middle, so when the full Moon is rising, the Sun is setting. Don't confuse this with eclipses. The Moon's orbit goes slightly above and below the Earth so with most full Moons, we are only mostly between the Sun and Moon, but its close enough.