How exactly is thermal energy transfered?

The hot rod will give some of its heat to the cold rod, this transferal of energy will probably happen by conduction. In solids I think the heat is a measure of the kinetic energy of the crystal lattice, the more heat the more the lattice shakes, the more kinetic energy it has. Metals are good conductors of heat, silicon is not so good; if you've got metal rods I think the heat will flow quite quickly from the hot rod into the cold rod across the boundary where they touch.

If there is a temperature gradient, heat will want to flow. It's like a hill, a ball will want to roll from top to bottom in much the same way that the heat will want to flow from high temp to low temp. So when the hot touches the cold there's a sharp gradient and a lot of heat is transferred.

 

Think about these two solids as two lattices both vibrating at different rates. The one with the higher temperture is vibrating faster. If they are touching than the faster lattice will bump into the slower hence cooler lattice and so the slower lattice will now vibrate faster but the faster lattice is slowered as it thinks there is more lattices to do the vibration against. In this way, they end up in an equilibrium.

If they do not touch than the faster lattice will vibrate against the air and the air molecules will have more kinetic energy so will randomly push against the slower lattice causing the slower lattice to increase its vibration. The faster lattice will over time vibrate slower due to transferring energy to the air molecules. So an equilibrium will also come but much slower than the first case.

In this way there is no tangible thing as heat just as there isn't for energy in general. Its a convenient quantity to do the bookkeeping.