How is it possible that heat may excite electrons even though it is a continuous energy source? How is it that electrons absorb thermal energy and become electronically excited, given that heat is a continuous source of energy, and electronic energy levels are quantized. I would like to reconcile those two ideas. 
 A: 
given that heat is a continuous source of energy,

Heat is kinetic energy stored in the vibrational and rotational degrees of freedom  of molecules and lattices in liquids and solids, and molecules and atoms in gases.
Heat is  a classical variable.

and electronic energy levels are quantized.

At the quantum mechanical level the atoms and molecules have spill over fields from the quantum mechanical shapes of the orbitals of the electrons .  Thus there are positive fields negative fields that set up potentials which create the attractive forces that hold molecules and lattices together.  In quantum mechanical terms these potentials exchange virtual photons while the molecules and the lattices are vibrating . 
When the temperature is high enough, and/or at the tail end of the vibrational kinetic energies, the energy needed for a transition of an electron from a lower energy level state to a higher state will become available.
A similar mechanism generates the real photons of black body radiation, from the vibrations and rotations of the atoms and molecules in the samples.

I would like to reconcile those two ideas. 

I hope this helps.
A: Heat described by classical physics is not discrete and can be described as the infrared region of the electromagnetic spectrum. One example is thermal imaging, which literally detects infrared frequencies. But with quantum mechanics (QM), you can see that electromagnetic waves can be thought as a bunch of photons with certain frequencies. So by QM, heat contains photons of energy with frequency lying in the infrared part of the spectrum. And now that we have the picture of photons we can use them to describe the excitation of electrons by thermal energy.
