From the cosmic perspective, 'mass' is energy in a state that, instead of traveling at the speed of light, wants to sit on one place (for reasons that were discovered later), and needs a push to get it moving. When you measure mass, you're really measuring how much you need to push a given amount of this inertially-fixed energy to impart a given velocity. Our ordinary units of energy, mass, and velocity came from observing these facts of physics before we figured out what mass and energy "are", and E=mc^2 ties the knot between the different units we came up with.

The really fascinating thing is that Einstein was able to propose E=mc^2 before any of that was known. It was just a funny term that showed up in the equations for relativistic energy and momentum, and somehow Einstein's intuition was that it might mean something more.