Why would a plume of solid silicate rock rising slowly from deep in the mantle begin melting as it neared the base of the lithosphere?

A plume of solid rock rising slowly from the deep in the mantle will begin to melt as it nears the base of the lithosphere because temperatures remain high as lowered pressures decrease melting temperatures.

For most substances, decreasing the pressure when a system is in equilibrium between liquid and solid phases will decrease the phase of transition temperature (temperature at which a substance changes from one crystal state to another).

This simply means the decrease in pressure causes a decrease in the temperature required to begin melting process.