The motor will spin much faster than it does normally when loaded by the pump moving water. This higher speed may wear out something much quicker than during normal operation.
The system should be able to handle the higher speed for a few seconds, but most likely it was not designed to operate at that speed long term. The extra robustness probably wasn't built into the friction points to be able to handle it, especially when there is no water flowing and thereby removing heat. Obvious failure points include the packing that seals around the shaft to keep the water out of the motor, the bearings, and possibly the motor brushes if it has any. The system may also not be balanced for the high speed, putting additional stress onto the bearings.
One thing that is NOT a likely failure cause is the motor overheating. Since it is spinning faster, it will draw less current, so the heating in the windings will be less. Unless the water flow is designed to cool the motor, this is not the issue. The ones I've seen simply have a motor sitting atop a pump, with no special water flow around the motor.
As with any device, when you violate the specs, you can't be surprised when it stops working, perhaps in spectacular ways. If the specs say the unit is intended to work only when water is present, then that's what it's intended to do. You can speculate on why that might be, but ultimately you don't really known. Running it without water is a Bad Idea.