How are you able to take a picture for 5 minutes while the camera is moving and not have the picture be blurry?

The stars are moving across the sky. More accurately, the Earth is rotating beneath the stars. The mount moves the camera very smoothly at the same speed as the sky appears to move from the Earth's surface so that the camera stays pointed at the same spot in the sky as the Earth rotates underneath it.

The pictures taken with the 5 minute exposures/mounts only compare to 5 minutes with no mount, and not 15-25 seconds no mount–how much better are these pictures and how so?

Longer exposures help in several ways:

• You can use a narrower aperture. This allows you to use the "sweet spot"of the lens you are using. Most lenses are softest and show the effects of aberrations the most when used at their widest aperture. Narrowing the aperture by two or three stops usually gives you the lens' best performance. It also allows you to avoid the need for very expensive, very wide aperture lenses.
• You can use a lower ISO setting. By collecting more light with a long exposure time you don't need to amplify the signal from the sensor as much. The problem with amplifying the signal too much as that you also amplify noise along with the signal (light). Less amplification is particularly helpful with reducing read noise which is in the same spot in every frame.
• You can take multiple exposures and combine (stack) them while covering the same part of the sky without repositioning the camera between each shot. This is another technique used to increase the signal to noise ratio by minimizing the random noise that appears in different spots in each frame.