I understand there is energy hidden in mass, but how does energy become mass?

With E=MC2, there is energy (huge amounts!) hidden in mass, in the form of atoms with their electrons, protons, and neutrons, right? But I can't understand how energy traveling at the speed of light becomes mass. Does the speeding energy take on more atoms? Do the atoms in the speeding energy swell? What happens that causes energy speeding at the speed of light to take on mass? Is it the way the energy moves? For instance, does the speeding energy vibrate and the vibration makes it SEEM like it's taken on mass? Help me understand this. It's making me crazy!