Why does light interact with mass? If light is just a disturbance in the electromagnetic field, then why does it interact with mass?
 A: Classically, because matter is made of charged electrons and protons. An electromagnetic field  or a disturbances in it exerts forces on charged particles. In particular, electrons are light and easy to push around.
Quantum mechanically, one talks the same thing with energy rather than forces. Forces change the trajectory of a particle. One of the reasons quantum mechanics is needed is that electrons do not behave like classical mechanics would lead you to expect. Electrons are described by a wave function. They don't have trajectories. There is a probability of finding the electron in any given position.
For a freely moving electron, it is possible that you might have something like a trajectory. The electron might probably be near position x now and position y later. But when an electron is bound to a nucleus, the probabilities don't change with time. The electron just stays near the nucleus. You can't say that it is moving or stationary. It has no trajectory.
It does have a probability of being found at any given position near the nucleus. There is a potential energy for each position based on the distance to the nucleus. So if the electron is found at some position, you know its potential energy. Given conservation of energy, you can also finds its kinetic energy.
When an electron is bound to nucleus, only certain energy levels are available to it.
Light is a disturbance in an electromagnetic field. These disturbances carry energy. Another reason quantum mechanics is needed is that these disturbances do not behave like classical electromagnetic theory would lead you to expect. A disturbance is an all or nothing thing, something like a particle. For this reason it is called a photon.
It is possible for a photon to ignore an electron in an atom. The photon passes on by without disturbing either itself or the electron.
If it does interact with the electron, it gives all of its energy to the electron. The electron jumps from one energy level to another. The photon is absorbed and disappears. No electromagnetic field disturbance is left.
It is also possible for an electron at a high energy level to spontaneously drop down to a low energy level. If it does, it generates a disturbance in the electromagnetic field. That is, it creates a photon, which flies away.
A: *

*Short answer:
Let's say that interacting means transferring energy and/or momentum between two systems.
Our systems are light and matter.
The interaction is possible because the light is an EM wave (both in classical and quantum schemes) that have this property: transports energy and momentum.

*

*Long answer:
Let's see how this works. A system is described by its Hamiltonian, our system is the light+matter. If light and matter interact the Hamiltonian will show it to us by a cross term between the wave and the "matter".
Classical (continuous energy)
The EM wave is the solution of a wave equation:
$$
-\vec{\nabla}^{2} \vec{A}(\vec{r}, t)+\frac{1}{c^{2}} \frac{\partial^{2} \vec{A}(\vec{r}, t)}{\partial t^{2}}=0
$$
$\vec{A}$ is the vector potential describing what you call "disturbance" of the EM field, that is, the EM wave.
It can be shown that a charged particle (obeying Lorentz force):
$$
\vec{F}=q(\vec{E}+\vec{v} \times \vec{B})
$$
have a classical Hamiltonian with the form:
$$
H =\frac{1}{2 m}[\vec{p}-q \vec{A}(\vec{r}, t)]^{2}
$$
that it's the classical Hamiltonian for a particle interacting with an EM wave.
You can go into detail about this topic in this link.
Quantum (discrete energy)
The "disturbance" in the EM field is still a wave, in this case represented as a set of independent quantized harmonic oscillators of energy:
$$
\hat{H}_{\mathrm{F}}=\sum_{k} \hbar \omega_{k}\left(\hat{a}_{k}^{\dagger}(t) \hat{a}_{k}(t)+\frac{1}{2}\right)
$$
This equation means that the photons (with definite energies ($\vec{P}=\hbar \vec{k}$)) are the normal modes of the EM field.
The total hamiltonian of the light and the "matter" have this form:
$$
\hat{H}=\hat{H}_{\mathrm{S}}+\hat{H}_{\mathrm{F}}+\hat{H}_{\mathrm{int}}
$$
Where $\hat{H}_{S}$ is the "matter" hamiltonian and $\hat{H}_{int}$ is the Hamiltonian interaction between the system and the field (the perturbation).
You can go into detail about this topic in the book: Quantum Optics for Beginners.
