Why don't eclipses occur during any other moon phases besides the new and full moon?

There's no eclipse at every full moon and new moon because the moon's orbit is inclined to Earth's orbit by about five degrees. Most of the time, the sun, Earth and moon don't line up precisely enough to cause an eclipse.

Explanation:

During a lunar eclipse, Earth comes between the Sun and the Moon, blocking the sunlight falling on the Moon. Earth's shadow covers all or part of the lunar surface.