How does the temperature of matter increase when it absorbs light? When matter absorbs light waves does it cause the matter to increase in temperature? For example, microwaves can heat matter up, so can light waves do the same?
 A: You are asking about light waves, now it is very important to understand that both microwaves and visible light are EM waves, but they do differ in energy levels they carry.
Now basically when a photon interacts with an atom, three things can happen:


*

*elastic scattering, the photon keeps its energy level, changes angle (mirror reflection)

*inelastic scattering, the photon transfers some of its energy to the atom, and changes angle (heating up deeper into the material)

*absorption, the photon transfers all of its energy to the atom, and the photon ceases to exist (heats up the material both at the surface or deeper into the material)
Usually when we talk about heating up an object by EM waves, we talk about absorption, and inelastic scattering.
Heating up a certain object can happen at the surface of the object or deeper into the material.

When light hits matter, it can be reflected, absorbed, raising the kinetic energy of the material, or for appropriate frequencies raise the energy levels of the lattice or the atoms. All these interactions end up in raising the temperature of the matter under study.
  depending on the frequency it can be kinetic energy, or vibrational or rotational in a lattice, raising the temperature.

Can light increase an object temperature?
It is very important to understand that the way heating up the material works depends on both the frequency of the photons and the type of material.
Microwaves
Now microwaves (I assume you are asking about microwave ovens) in your example are tuned to the frequency of water molecules in food and these EM waves are able to enter deeper into the food and transfer their energy to the atoms/molecules inside the food and thus giving kinetic energy to these atom/molecules, thus heating up food.
Now certain materials just let these microwaves pass through them with minimal heating up (plastic, glass, ceramics), others reflect them (certain metals).

The microwaves are primarily designed to vibrate/heat the water molecules in the food.
  Microwaves that cook your food pass through plastics, glass, and ceramics, with mimimal heating.
  Metals, on the other hand, reflect these radio waves.

Why does the food in the microwave heat up but the bowl doesn't?
Visible light
Now you are not saying it explicitly, but I assume you are asking about visible light too. Visible light photons have energy levels so that they heat up certain materials (most opaque solids) around us on the surface, being absorbed by surface atoms/molecules, transferring their energy to the atoms/molecules kinetic energy thus heating the material up. Other materials (water and other translucent materials) are being heated up by visible light as photons pass through them, some of these photons get absorbed or inelastically scattered, transferring energy to the atoms/molecules. Other materials (mostly reflective, like shiny metals) get barely heated up by visible light, because most photons get elastically scattered (reflected). It is a common misunderstanding, but Sunlight does contain non-visible light, and when Sunlight shines on metals, most of the heating up of the metal comes from non-visible wavelengths.
Other wavelengths
Now there are other wavelength EM waves, other then microwave and visible, and they all interact differently with different materials, depending in the material and the wavelength of the EM wave, but the main types of interactions that lead to heating up are the same, absorption and inelastic scattering.
A: When light falls on the matter, it's called light-matter interaction. But, what you are taking is molecular vibration. 
Matters are made up of molecules and they stick together by what we call chemical bonds which are nothing but electromagnetic interaction. Light is nothing but electromagnetic waves and wave have a wavelength which corresponds to energy. Molecules love to jiggle along with these waves which eventually increases their energy and hence the temperature.
