What is the rubber hose hanging from rear axle of a Ford Explorer?

That's most likely the ABS (Anti-lock Braking System) sensor wiring harness. While it might look like a hose, it's actually a protected wiring harness that runs from the ABS sensor on the wheel hub to the vehicle's ABS module. It's often shielded with a rubberized or plastic outer covering to protect the wires from damage.