Explain, in terms of particles, how heat is transferred through the glass wall of a boiling tube

Considering the great topic of thermodynamics (assuming normal atmospheric pressure):

Keep in mind that hotness (temperature) and heat are related but not the same thing. Each time water goes from one state to another, it requires the gain or loss of heat.

In this case the boiling water is gaining heat, the water molecules are vibrating faster with more kinetic energy (energy of motion). The (heat) energy is transferred by the rapid movement of particles colliding with other particles.