How does kinetic energy not violate the Law of Conservation of Energy?

It is important to distinguish between invariant quantities and conserved quantities.  Energy is a conserved quantity. It is conserved in every inertial frame. It is the time component of the momentum 4-vector, so it is not invariant. It is not the same in all frames.

I would like to clarify Sahal's answer. A conserved quantity is the quantity that does not change over time. An invariant quantity is the quantity that does not depend on the frame of reference. Examples: Conserved quantities: energy, momentum. Invariant quantities: invariant mass, space-time length, Minkowski dot product of two relativistic 4-vectors. When you switch the frame of reference, energy and momentum, which are non-invariant, change; however, they remain conserved within each frame of reference.

Thanks for the A2A As others have pointed out energy conservation is frame independent and if one has to apply it across frames then both such frames need to be considered with respect to another frame of reference such that the energy possessed by the two 'frames' themselves is taken into account. Take you own example: when you begin to move at the speed of the body, you yourself acquire kinetic energy with respect to 'ground' frame or another way to look at it is that apart from you and the body, the rest of the universe just gained kinetic energy by virtue of its motion with respect to you. It is important to note that while energy is always conserved, it is not always seen in the same form in different frames. What may be perceived as kinetic energy in one frame can be shown a potential energy in another case(as happens in your example).

No it doesn't violate. Nobody violates the law. When you're moving with the object, you too possess kinetic energy. Since you're taking a reference point travelling at same speed, the object SEEMS stationary. It isn't stationary. It always possesses kinetic energy. While standing, you possess a lot of kinetic energy because you are travelling in space at high speed on earth. But in calculations, we consider you at rest as you are not moving with reference to another steady object on earth. So, we assume you don't possess kinetic energy. The law is the law.

If u  yourself are moving,then yours is a non-inertial frame,and u have to consider pseudo force acting on the object.This force will then indicate the kinetic energy that was "lost" when ur frame started moving

In Newtonian mechanics, kinetic energy is reference frame dependent, however by the relativistic approach, kinetic energy is independent of the reference frame. In the Newtonian approach,  it cannot be said that the system does not possess kinetic energy.  The system does possess energy when viewed from a certain stationary frame. Conservation of energy is not being violated. Because in another frame the system has zero relative velocity does not mean that it has zero energy.

the term "energy" applies to many different things. Measured values depend on the farm of reference. so far it seems that energy can change forms but there is some "thing" in the universe we refer to as Mass_Energy..We ASSUME the total is constant so we define new ways to describe "it " whatever "it " is!