why do the tones sound full and rich when you sing in the shower?

It is called reverberation.

See:
http://www.wisegeek.com/what-is-a-reverberation.htm#didyouknowout

When you sing in the shower, you may notice that the tones sound full and rich. This phenomenon can be attributed to a few factors:

1. Acoustic Reflection: Showers are typically enclosed spaces with hard, reflective surfaces such as tiles, glass, and ceramic. These surfaces bounce sound waves around the shower, causing multiple reflections. These reflections reinforce and amplify certain frequencies, making the sound appear fuller and richer.

2. Reverberation: The combination of hard surfaces in a shower creates a longer reverberation time. Reverberation is the persistence of sound after the initial source has stopped. This prolonged reverberation adds depth and resonance to the tones you produce while singing, enhancing the overall sound quality.

3. Enclosed Space: Compared to singing in an open area, like a room, singing in the enclosed space of a shower creates a more contained sound environment. This containment prevents the sound waves from dispersing too widely and allows them to interact more closely with each other, resulting in a stronger and harmonically rich sound.

So, to summarize, the full and rich sound when singing in the shower is due to the acoustic reflection, longer reverberation, and the enclosed space, all of which contribute to enhancing the tones you produce.