The concept behind a solid state nonvolatile memory system is pretty simple. You must store data in an array of solid state components and maintain that sata even when power is removed. How stuff works has a good beginers primer on the technology at hxxp://computer.howstuffworks.com/flash-memory.htm
We could use solid state storage for everything, but considering that flash ram costs about $50 a GB, and magnetic disc space costs less than $1 per GB its easy to understand why we don't yet. As Flash prices drop watch for more solid state storage devices.
Space is really not the issue, microdrive technologies allow us to store information on very small magnetic discs.
As for why solid state is shock proof(for the most part) its easier to explain why a hard disk isn't. A hard disk has a magnetic platter with a small read/write head mounted just above the surface. Ideally there should never be contact between these two, since the platter is easily damaged. When you drop/shake the drive you cause the head to touch the platter and in most cases permanently damage that section of the drive. Because flash ram has no moving parts theres nothing to bang together so the only way to damage it through physical shock would involve using enough force to break the actual chip.
The schematics for a player are pretty boring, the typical commercial unit uses a single IC MP3 decoder, a flash controller/memory, and a USB communications chip. A basic microcontroller makes it all work together.