Why does the kinetic energy of particles increase on heating? One of the obvious facts in thermodynamics is, when we provide heat to a system, some of the heat energy get stored into the internal energy of the system. It is thought that it was stored either in form of  potential energy or kinetic energy, but here we give more emphasis to the kinetic energy term, because it is in the core of the theoretical framework of kinetic theory of gases. It is very easy to experimentally confirm this fact, but what is theoretical reasoning behind it?
More specifically, why and how does the kinetic energy of the particles of a system increase when heat is provided to the system?
 A: There are several mechanisms which can serve to increase the kinetic energy of a system.  Any of these might be ways in which heat / energy is added to the kinetic rather than potential energy.  I'll give a few examples of different energy sources and the conditions under which the added energy might preferentially increase the kinetic- or potential-energy terms:

*

*If you have a beaker full of gas sitting in direct contact with a hot object (like a hot plate), the molecules of the beaker will occasionally collide with those of the hot object. Collisions tend to evenly divide the kinetic energy between the participating molecules / atoms.  Thus energy is transferred from the hot object to the cold one. This serves to cool the hotter object and warm the cooler one.  If you can arrange a system so that the molecules don't directly interact, or don't redistribute energy on impact, you can insulate the hot and cold bodies from each other.  The most effective insulator is a vacuum - i.e. preventing the bodies from directly touching.

*If you shine a bright light - or microwave oven, or laser, or maser or incandescent light, or other strong EM energy source - onto or through an object, the photons have an opportunity to interact with the electrons in the object.  If they don't interact with the object, they pass through it without reflection or absorption and the object is clear at that wavelength. If the incoming photons have wavelengths / energies which happen to be resonant with allowed transitions in the atoms / molecules they hit, some of the photons' energy will be absorbed into the atoms. The details depend on which transitions are stimulated by the radiation, but generally you can either raise electrons' potential energy on absorption or add energy to rotational and vibrational states.  Rotational and vibrational energy are directly components of kinetic energy, so immediately increase the temperature of the absorbing substance. Energy absorbed into electronic energy states may either be re-emitted or get redistributed into rotational/vibrational modes, depending on the structure of the absorbing molecules, how much they interact with neighboring atoms, and overlap between the electronic and rovibrational transitions.  If the incident light is re-emitted immediately at the same wavelength, it's reflected. Metals are shiny because they have large numbers of free electrons that efficiently re-emit incident light at many wavelengths.  If all the light is reflected or transmitted without being absorbed or redistributed among molecular motions, no heating occurs.  The details of what happens (absorption vs reflection vs transmission and which mechanical mode is stimulated) are going to depend on things like the density of "free" electrons, the polarizability of the atoms' / molecules' electron clouds, the frequency of the incident light, the masses of the atoms, the polarization of the incident photons, etc.  However, the core physics involved is that an EM wave ("light" / "photon") contains an oscillating electric field (and an oscillating magnetic field, but usually the electric component is more important).  When it hits a molecule, the electrons and protons feel opposing forces due to the electric field. This sets up an electric dipole in the individual atoms, and that dipole then feels a force in response to the still-present, still-changing electromagnetic wave. The electrons have much lower masses than the nuclei, so respond more quickly, but still not instantaneously - resulting in the electron oscillations having larger amplitude than the atoms. (An electron cloud also surrounds each nucleus, which tends to shield the nucleus from the direct interaction with the photons.) The forces binding the electrons to the nuclei, and the forces binding together different atoms in a molecule, and the forces binding different molecules in a medium then serve to distribute the induced electron oscillations into other mechanical modes... as the energy imparted to the electrons by the photons leaks into other modes of motion, it becomes less like a "stimulated oscillation" and more like "thermal energy".

*Particle radiation (like neutrons, alpha particles, or electrons / beta particles) themselves have kinetic energy.  When those particles are trapped / absorbed by an object, the particles' energy gets distributed among the particles it impacts. This is what causes heating in nuclear reactors or when radiation therapy is used to cook a tumor.  Materials which do not efficiently absorb the given type of radiation are heated less by it.

*If you compress a gas, you're directly increasing its potential energy.  By the virial theorem, that increased potential energy tends to get redistributed into kinetic energy.  As a result, an increase in temperature accompanies the increased pressure.  If you play with the details of the compression and expansion, and with the details of how the compressed gas interacts thermally with its environment, you can build either an engine or a refrigerator.

In summary, and to directly answer your question, the details of "why and how kinetic energy of the particles of a system" increase when heat is provided depend on how that heat energy is absorbed into the system.
A: When we add heat to matter its energy increases, both kinetic and potential.  How much of the energy we add goes to kinetic versus potential depends on the details of molecular interactions. In the ideal gas state there is no interaction, therefore all heat goes into kinetic energy. When molecules interact, some heat goes into increasing the potential energy as well, which is to say, molecules can spend more time in configurations that have high potential energy. This is what happens when a liquid, whose molecules are mostly trapped in regions of negative potential due to attraction, turns into a gas, whose molecules are mostly at nearly zero (and thus higher) potential.
We talk more about kinetic than potential energy because it is mathematically simpler to deal with. The total kinetic energy is the sum of kinetic energies of all particles, a simple additive formula. Potential energy on the other hand is much harder to handle because it depends on the arrangement of molecules in space. Move one molecule and the potential energy of all molecules changes in a way that cannot be calculated by a simple sum.  The ideal gas is so simple to describe mathematically precisely because it has no potential energy.
A: The energy of a system of particles, interacting via pairwise interactions, can be written as
$$
U(\mathbf{x}_1,...,\mathbf{x}_N; \mathbf{p}_1,...,\mathbf{p}_N) =
\sum_{i=1}^N\frac{\mathbf{p}_i}{2m_i}+
\frac{1}{2}\sum_{i,j=1}^NU(\mathbf{x}_i-\mathbf{x}_j)
$$
If the total value of this energy were to increase, this increase has to be distributed between the kinetic and the potential energy. How exactly it is distributed, depends on the system configuration in space and momentum space. However, the potential energy typically decreases with the distance between particles - in particular, in gases this contribution is very small, which is pretty much the definition of gas (as opposed to more dense liquids and solids, whose properties are determined by the interactions.) In particular, ideal gas is by definition is a gas with no interactions whatsoever (although some residual interaction is assumed, necessary for establishing thermodynamic equilibrium.)
The general theoretical framework develops a theory for gases, because many results for gases can be obtained in exact or nearly exact form. It then develops various approximate methods for dealing with other systems, like liquids and solids, and the transitions between different phases, and a big part of physics community is actually busy applying these methods to calculate the properties of these systems. (The methods range from ranging from virial expansion covered in basic stat physics textbooks to more advanced stuff like renormalization group and quantum field theory methods for statistical physics.)
How exactly the energy transferred to the system is distributed through it depends on the nature of the system. The power of the thermodynamic arguments (the part of the stat physics texts without equations) is precisely that we can say a lot about the system properties without getting into these system specific mechanisms of energy redistribution.
A: If we add energy to a system via heat or work, it doesn't matter, we increase its internal energy. That is the content of the first principle of thermodynamics. A point that may not be obvious is how such an increase in energy affects the average potential and kinetic energy separately.
Classical Thermodynamics may provide a firm guide to discussing such an issue, although it cannot shed light on the microscopic mechanism behind the phenomenon. Still, it is helpful to start from here. Thermodynamics says that, for a stable thermodynamic system, the temperature must be an increasing function of the energy. It is a property of thermodynamic systems directly connected to the positive definiteness of specific heat. Notice that the case of no increase in temperature when energy increases (at a first-order phase transition with latent heat) is included.
Therefore, the modified question should be: why does the kinetic energy of particles increase on heating but remains constant at a first-order phase transition?
Statistical Mechanics provides the answer.
Firstly, Statistical Mechanics allows for establishing a link between temperature and average kinetic energy of the system for classical systems (equipartition theorem). Moreover, it allows justifying the positiveness of specific heat, including the case of diverging specific heat at phase coexistence.
Of course, Statistical Mechanics does not describe the microscopic mechanisms, but it helps to understand the critical ingredients for the observed behavior. Indeed, it shows that such a  behavior is shared by every system described by Hamiltonians (aka interactions), allowing the so-called thermodynamic limit. The case of gravitational interaction is instructive: they are systems where the usual thermodynamic limit does not exist and are characterized by a negative specific heat: their temperature increases when energy decreases. It is a trivial consequence of the viral theorem. Such a finding establishes that the usual behavior of laboratory thermodynamic systems is a property of the particular class of interactions corresponding to Hamiltonians amenable to the thermodynamic limit.
We can conclude that the non-decrease of the average kinetic energy is rooted in the way the number of microscopic states increases with the system's energy. Only such a property decides how the added energy is partitioned into kinetic and potential energy.
