How does the mass of ice cubes affect its melting time?

Yes.

Keep in mind that it takes one BTU to reduce or increase the temperature of one cm3 water by one degree celsius.

Per physics, cold moves toward hot, so the temperature of the ice can be said to move into the temperature of the water. (This can help you visualize the process).

As the ice melts, it cools the water around it. The greater the volume, the longer the cooling process takes. This is also affected by the surface area and the amount of it in contact with the water.

To break it down simply, the greater the volume of water, the faster the ice will melt.