How would you use the phase diagram of water to explain why ice at the bottom of a glacier can melt when the rest of the glacier which is the same temperature remains solid?

When something changes its state of matter (formally, a phase change), regardless of whatever it is, it remains at it's melting point/boiling point/freezing point UNTIL all of it has changed its state of matter.

So, to clear up, ice and water can both exist at 0°C at 1 atmosphere. The difference is that water has more energy than ice does, therefore solid water can break up the bonds and become a liquid.

I heavily suggest you take a look at calorimetry, it explains it greatly and is related to your question. :)

Now here pressure is plotted against temperature, and the transition between liquid and solid and gas is mapped out....And reasonably at high pressure and low temperature, we gots a solid phase ice, and at high temperature and low pressure we gots a gas....(i.e. water vapour).

Now look at the interface between ice and water.....the slope is NEGATIVE...

And now we introduce the Clapeyron equation for phase equilibria....

#(dP)/(dT)=(DeltaS)/(DeltaV)#

Looking at the slope of the transition between ice and water, we gots a NEGATIVE slope (in fact the slope is exaggerated somewhat...)....and this is clearly the differential #(dP)/(dT)#...and here this differential is NEGATIVE....

And since #(dP)/(dT)=(DeltaS)/(DeltaV)#....we go from SOLID to LIQUID, and clearly the statistical probability for disorder, the entropy, is a POSITIVE quantity, this means that #DeltaV#, the change in volume, must be NEGATIVE.....i.e. solid water occupies a GREATER volume than liquid water....with the result that ice bergs float....ice/water is unusual stuff, for most other materials, the liquid state is LESS dense than the solid state.

And so finally to your question.....under a regime of HIGH PRESSURE, i.e. at the bottom of the glacier, the melting point of the ice is REDUCED with respect to its melting point at #1*atm#...as the graph clearly shows....