The answer is in the ideal gas equation stated in your question: Liquid propane is in equilibrium with propane vapor inside the tank. When vapor flows through an open valve, the drop in pressure (at relatively constant volume) must be balanced by a proportionate drop in temperature. Alternatively, think of the molecules of propane that escape the tank through the pipes: they have higher kinetic energy than the rest of the propane molecules left behind in the tank. Temperature is average kinetic energy, so the tank cools as the most energetic molecules leave.
The same thing happens with those cans of compressed air dust sprays that use a liquid propellant with a high vapor pressure—just like the propane in your tank.