The c squared term basically comes from the kinetic energy of an object in motion that is for a body of mass m moving at a velocity of v the kinetic energy increases as v squared now to sort out the squared term.

The measure of energy consists of two parts, an applied force and a distance moved.

So when adding energy to a body by applying a force for a particular time period we need to consider how big the force is and how far the mass would move

The most fundamental equation of motion is that the acceleration a mass is proportional to the force applied

Next we have to find out the distance traveled in a time period as a result of the acceleration that the force has created

for a uniform acceleration from stationary the velocity is the aceleration times the time over which it has operated

and the distance covered is the velocity times the time period too

so for a given time period the distance travelled is a constant (1/2) x velocity x velocity

That is, vlocity comes into it twice to make it velocity squared