What actually happens when a photon is absorbed by matter? In my physics courses so far we've only discussed the before and after of a photon being absorbed by matter. But what actually happens here? How does the light "meld" with the atom that it is incident upon? What exactly is the mechanism of interaction between matter and the incident photon? 
 A: Particles are represented by fluctuations in fields in quantum field theory. So if you have a photon and an electron you will have a corresponding fluctuation in each field. The two fields interact with each other and so the fluctuation in the photon field can influence the fluctuation in the electron field and disappear. 
If you want an analogy think of two pieces of string tied together at one end. There is initially a fluctuation in both strings which collide at the string interface. After the collision there is just one larger fluctuation in the string representing the electron field.
A: Consider the thing within second quantization form: the Hamiltonian contains term like $b^\dagger a_e+\text{h.c}$, which describes the annihilation of a excitation would produce an photon, and its conjugate procedure (which you cares more): absorbing a photon and becoming excited. 
This kind of Hamiltonian is derived, in principally, from QuantumElectroDynamics (QED). The Lagrangian contains such a vertex term: matter field (spin-$\frac{1}{2}$ field) couples to light field (spin-$1$ field). Wikipedia gives a clear enough explanation for this. 
A: There are many ways a photon can interact with matter, but since the photon is a quantum mechanical entity, one has to define matter in the quantum regime.
In dimensions commensurate to h_bar  matter is composed of atoms in various combinations.
One way of interacting with matter can be seen here: Atoms are modeled by electrons in orbitals around a positive nucleus of equal and opposite charges. The orbitals are solutions of the quantum mechanical equation and are energy levels. So a photon can be absorbed by an atom by "kicking" an electron to a higher energy level , if the photon has that energy +/- a quantum mechanical width in the energy. ( the photon's energy is h*nu, where nu is the frequency of the classical light beam it is a part of). This gives absorption spectra, and then when the electron falls back to the lower energy state emission spectra. Momentum is conserved by an excess in motion of the atom that absorbed the photon.
The atoms in matter have also a spill over electric field with which a photon can interact with the Compton effect, either with single atoms or with a mass of atoms as in a gas, a liquid or a solid. The photon loses part of  its energy and momentum and transfers it to the atoms or molecules , depending on the scatter the energy can be vibrational energy in a solid or liquid, thus in the end the energy heats matter up, because temperature is connected with the average kinetic energy of atoms and molecules.
There are other more complicated ways of the photon scattering  and giving up part of its energy and sequentially be completely absorbed.
A: The mechanism of interaction is very much the same as a radio wave interacting with an antenna. The "photon" is manifested as oscillations of the electric field, which drive the electron like a mass on a spring. The oscillation frequency is given by the difference between the initial state and the excited state, and you can track the oscillating charge motion calculating the superposition of the two states.
The oscillating electron radiates electromagnetic energy just like any other antenna. Initially, because of the interaction of the radiated field with the incident field, there is actually a net absorption of energy; but eventually the incident field goes away and the atomic system then simply continues to oscillate, re-radiating any residual energy until it returns to the ground state.
You can also analyze this system in terms of "photons" using something called "Fermi's Golden Rule", but it all comes to exactly the same result in terms of what you can actually measure...amount of scattered radiation as a function of the incident field.
DISCLAIMER: I am a recognized crackpot whose opinions are routinely and massively downvoted by the experts in this forum who know much more than me.
A: I will try in a very general sense, and then you can use your imagination, because books may not answer this for you.
Every interaction takes place via some kind of force. When we push a car, we transfer our energy into the car, but the energy first coverts to force, and then goes into the car. Same way, the photon must transform into a tiny force which would push the electron. That force more than likely has to be electromagnetic repulsion. You can think that the photon field/wave of certain frequency, and field/wave of impacted electron, can not co-exist in space. One transfers its energy to the other, and disappears. This non-ability to co-exist gives rise to repulsion, repulsion gives rise to movement. Paradoxically, they still co-exist post interaction, but as one, not as two.
