How does a speaker blow out?

What Erik said (regarding clipping), but: after a x-over passes that high freq distortion to the tweeter and burns it (typically: open), the energy that no longer has a path that way, is routed to the next highest freq driver, in the circuit.

True, but since most crossovers are in parallel, the HF voltage seen by the woofer remains the same. More likely to burn the first coil in the low-pass filter than the woofer itself, but things happen. :)

The amp having less current draw due to an open tweeter circuit could mean a higher voltage though. 

Well, the theory is actually that the clipping causes a square wave, which has a lot of high frequency harmonics, not DC exactly, so more likely to blow a tweeter.

It can happen for a lot of reasons including age and prior abuse.  There are two kind of failures.  Mechanical and electrical.

Mechanical means that the surround or the spider has failed, either from age or excess physical motion.  The electrical failures are caused by the melting of wires in the voice coil or to/from it or a disconnect.