What is a generator? A generator is the object that tells you how something changes under an infinitesimal symmetry transformation. We need to entirely understand this via the lens of Lie Group and Lie Algebra. generator lives in the tangent vector space and is making the rotation on infinitesimal scale.

The laws of nature are the same at all times. That means time translation is a symmetry. So there must exist a generator H such that:

This H is the energy, generator of time change or translation and still keeps the symmetry/invariation. Actually the Schrödinger equation can be derived just based on this fundamental understanding of symmetry and generator concept.

We can apply Noether’s theory to infer that time symmetry points to a conserved quantity, and it is the generator: energy.

From the 4-component space time view point, we say Energy generates temporal translations, Momentum generates spatial translations, position generates momentum translations and Angular momentum generates angle translations.

To think deeper, energy is the measure of time change or rate of temporal transformation, so the higher the change rate, the more energy.