Why do stars appear as points of light? What do you think happens to the light from a star that is moving...?

All the stars are very far away from the Sun, so their apparent diameter is very close to zero. No telescope on Earth has ever been able to resolve a star into a disk, although the Hubble Space telescope has done this for one or two stars, notably Betelgeuse in the constellation Orion. The light waves in a star moving towards us are compressed, just as sound waves from an object moving towards us on Earth are. This is known as the Doppler effect.

They appear as points of light because all of them except for the Sun are many light years away. We cannot really see the surfaces of distant stars.

The light from stars moving toward us will appear more blue than it actually is because of a type of Doppler shifting. It's called a blueshift.