Question

Why does the moon appears to be different sizes

Answer 1

Because of the position of the Earth. When we spin around on our axis, the moon turns around too, because of shadows, the shadow covers most of the moon at certain times

Answer 2

The Moon actually stays about the same apparent size no matter where in the sky it is. It simply looks larger when near the horizon because of a trick of perspective. When near the horizon, you have objects along the skyline to provide a reference to compare the Moon to. 
Use a quarter held out at arm's length to gauge against the Moon when it is low in the sky, and later on when it is higher up. You will find that the Moon is the same size relative to the quarter in both instances. 
If you want to get technical, the Moon actually appears smaller when it is near the horizon, because as the Earth rotates, your location on the Earth's surface gets closest to the Moon (when it is directly overhead). However, the difference in size is so small (about 1%) that it is not noticeable to the naked eye.