Energy is not a physical property. It is an abstract mathematical quantity that turns out to be very useful to calculate because it tells you important information about the state of the system. There are many kinds of energies associated with motion (kinetic energy), position (potential energy), mass ($E=mc^2$), and others. When physicists speak of energy transferring energy from one system to another, it is shorthand for talking about some physical processes changing the state of two systems. In many interacting systems, Newton's Third Law states that interactions have opposite effects on the two systems: one speeds up, the other slows down. The overall effect is to keep the total enegy associated with motion constant. In fact, physicist often choose a definition for energy so that it stays constant to make calculations easier. Different kinds of energy can be converted into others as well.

As an analogy, think of profit in business. I buy a car for \$5,000 and sell it for \$7,000. My profit is \$2,000. But, where was the amount of \$2,000 ever involved in the transactions? It wasn't. Profit is an abstract quantity of money that is useful to calculate since it tells you important information about the state of your business.

As for mass changing with energy, this is a subtle question. A moving object does not gain mass from kinetic energy. Heating an object does. You can think of mass as the total energy of a system as measured by an observer that is not moving with respect to that object. If the ovject is moving, the energy due to motion has to be subtracted away to get the mass.