I feel like this was a conversation of justifying why Zod is able to keep up with Clark and less how Kryptonians do what they do as originally asked.

In answer to that, Kryptonian biology allows their cells to act as batteries specifically to radiation. Different types of radiation have different effects on Kryptonians with the Yellow Star's radiation producing the best effects while the radiation of Krypton's core produces the worst as it can cause death to a Kryptonian. 

A Kryptonian's invulnerability comes from an enhanced bio-electric field which essentially acts as an extra layer of skin. This field can also extend to their clothes or anyone they come in contact with. This is why Clark can fly with Lois and Lois remains unharmed, it also explains how Clark doesn't hurt Lois if he were to catch her at a speed that rightfully should kill her. The Kryptonian's ability to shoot laser beams comes from stored solar energy expelled through their optical lenses which focuses the energy. They can adjust the focus level allowing for microscopic precision with their other optical abilities. This 'Comic Book Science' extends to all of the Kryptonian's abilities as the stored solar energy boosts their physical capabilities. It should also be noted that among Clark's lesser known abilities are a photographic memory, omni-linguism, and super senses.