Why do electrons absorb and re-emit photons?

And electrons in an atom absorb and re-emit it. But why do electrons bother to absorb and re-emit light and not just let it pass all the time?

There is a basic misunderstanding in your question.

An electron is an elementary particle of fixed mass. It can scatter off a photon, (which is also an elementary particle); if accelerated it can emit a photon, but it does not absorb it, because the electron's mass is fixed, and if it were able to absorb a photon - at the electron's center of mass - the mass would have to change, which contradicts observations and special relativity for elementary particles.

The terms absorption and absorbs are not usable with free electrons. It is the bound electrons in an atomic system, which may change energy levels in the atom when the atom absorbs a photon. So it is not the electron that absorbs the photon, but the atom.

The atom has energy levels, and if the photon energy coincides (within a small $ΔE$, the width of the energy level) with the transition energy of kicking an electron to an empty energy level, then the atom can absorb the photon (not the electron). So the answer to "why", above, is "because the photon has the appropriate energy to transfer the electron to an empty energy level".

If the photon energy does not coincide with transition energy of the atom, the photon may scatter with the spillover electric fields of the atom or molecule either elastically, or transferring energy and a lower energy photon continues on its way.

The relevant thought to keep is that an elementary particle cannot absorb a photon. Composite ones as atoms, molecules and lattices, can.

But why do electrons bother to absorb and re-emit light and not just let it pass all the time? (An electron would also be unstable by absorbing the energy and thus it re-emits it but in the first place why does it absorb it?)

A similar question could be asked about macro objects, say, a pendulum.

If you push a pendulum it is going to up and then it goes down. So, why, you could ask, does it bother to go up, if it is going down afterwards? Why does it absorb the energy of a push instead of just ignoring it?

I guess a simplistic answer is that it absorbs the energy because it gets a direct hit and it's not up to the pendulum to decide whether it should take it or just ignore it.

It's really down to two questions: why do electrons interact with photons, and why do atoms absorb photons?

Why interact with photons?

One can understand why electrons interact with photons by considering relativistic quantum field theory. In order to combine quantum mechanics with special relativity, you have to think of reality as consisting of "quantum fields". A field is something that has a value at every location, for example $\Phi(x,t)$ might be a (time-dependent) field, the value of the function signifying the value at every point in space (and every time t). A classical, non-quantum, field simply has a value at every location - you can think of it as the height of some system, say the deviation from equilibrium of an oscillator, at every point in space. A quantum field instead has a quantum system at every point in space; you can think of it as having a quantum harmonic oscillator at every point in space. The state of the point-like system, i.e. the deviation of this oscillator from equilibrium, is the "height" of the field at that point in space.

Now a core principle of quantum mechanics is that the phase of the quantum state does not matter. In order to carry this principle into a quantum field, the equations describing the physics of the system, known as the Lagrangian, has to not change if we change the phases of the states of the points in space. This requirement is known as "gauge symmetry". Now it so happens that it's rather difficult to build a gauge-symmetric Lagrangian using only standard expressions like derivatives. Instead, in order to maintain gauge-symmetry one has to introduce another quantum field, known as the gauge-field. This is the only way to maintain gauge symmetry, i.e. to maintain the requirement that the phase of a quantum state has no physical meaning.

So if you try to build laws of physics (a Lagrangian) to describe a simple matter field (e.g. an electron's field), you need to introduce an additional "gauge" field that interacts with it. The waves in the matter field will be the matter particles, such as electrons. The waves in the gauge field will be force-carrying particles, such as photons.

To summarize then, the reason an electron interacts with photons is that an electron is really a wave in a quantum (relativistic) field, and these waves have to interact with waves in the (gauge) electromagnetic field, which we call photons, in order for the electron's field to be a quantum field (i.e. for the phase of the point-like states to lack any physical meaning).

Why do atoms absorb photons?

Anna v beautifully explained already why an elementary electron cannot absorb a photon - it has to scatter it instead, as the electron's energy and hence mass cannot increase in its rest frame. But why is it that atoms absorb photons?

The important point here is that you cannot turn the electromagnetic interaction "off" for one effect while keeping in "on" for another. If you build an equation describing an electron that's attracted to a positive nucleus by the electromagnetic force, then this same system will also be affected by waves in the electromagnetic field.

So the same equations that describe the stable orbits (the electron levels/orbitals) due to the electromagnetic interaction with the potential energy of the nucleus, also describe a response to an electromagnetic wave (usually dealt with only as a perturbation off the stable state). And this interaction with the waves amounts to annihilating a normal-mode of the wave (annihilating a photon), while at the same time increasing in energy to maintain energy conservation. (Or conversely creating a normal-mode wave while dropping in energy.)