The equation E = mc², made famous by Einstein, contains an important clue as to the true nature of energy.
First thing to note is that the equation does not distinguish between different types of energy. It does not matter if the energy in question is kinetic or potential. All energies have the exact same effect on matter.
An increase in energy results in an increase in mass, regardless of energy type.
The natural conclusion to draw from this is that all energies, regardless of type, reside inside the particles that make up light and matter. If mass is a property of matter, and mass and energy is equivalent, then energy too must be a property of matter.
Next thing to note is the relationship to the speed of light, expressed by the constant c.
If inertial mass is a measure of time delay in energy distribution within particles, and time is a relationship between the speed of light and the size of particles, then there must be a relationship between energy, inertial mass and the speed of light.
This is exactly what the equation E = mc² is telling us.
Subatomic particles: neutrino, photon, electron, proton
Finally, if energy is stored as size of subatomic particles, so that an increase in energy is equivalent to an increase in size, then there is no mystery as to why an increase in energy will lead to more inertia.
The increased size of subatomic particles, due to an increase in energy, results in more time required for further increase in energy to take place.
The principle of mass-energy equivalence, as expressed by the equation E = mc², is fully compatible with the idea that energy is stored as size in subatomic particles.