Simply put, it is just idiomatic to say that. "On" can be used this way, to refer to the part of your body that is in contact with a surface:
- "I've been on my feet all day"
- "He is standing on his head"
- "He is lying on his back".
- "I fell and landed on my nose".
The statement that you are "lying on the bed" is about your location, whereas "lying on your back" is about your body's position.
It might also be worth noting that, idiomatically, people sleep in a bed, not usually on a bed. The first refers to getting under the covers, and when people do say "on" a bed it usually means there are no covers, or that the subject is lying on top of them.
In a similar way, we say people swim, or float on their back or front; and likewise, "in the water" and "on the water" would imply different things.