Why does energy impart the ability to do work?

I want to address this:

Like what it is about energy that it can give electrons the ability to move perennially around the nucleus?

It is not energy that keeps the electron at an energy level, but quantum mechanics, which mathematically imposes fixed energy levels. The electron in an orbital around an atom does not change energy , unless interacting with other particles. Then energy levels can be changed by the energy carried out or brought in by other particles

What it is about energy that it can allow the enzyme to move around in cells and replicate DNA etc?

A more complicated form of the argument above, since enzymes and dna are at the quantum mechanical microscopic frame. Energy levels and exchange of energy through interactions

This question, as asked, is a little circular. Energy is defined as the ability to do work. So literally you are asking “Why does the ability to do work impart the ability to do work?”

Energy isn’t a separate thing, it’s own independent entity. It is a way of describing a system. The laws of physics permit some systems to exert a force over a distance. Such systems are described as having energy. The system can do work by the laws of physics governing the system, and the amount of work the system can do is its energy.

To help conceptualize energy it is useful to think of a spring. (Many of the laws of physics resemble springs in some important ways). A spring follows Hooke’s law which says that a spring has an equilibrium length and as you stretch it or compress it the spring produces a force pushing back towards equilibrium. When a spring is stretched Hooke’s law imparts the ability to exert a force over a distance. As such, the spring is described as having energy. It is Hooke’s law that provides the ability to do work, energy simply describes and quantifies that fact.

One abstraction that has helped me understand energy(E) is: change. So the ability to do work, is also energy. Water stored in a reservoir has potential energy, the ability to do work on a turbine, and different types of work can be quantified and measured: mechanical, chemical, thermodynamic (heat), hydrodynamic, etc. Willing to bet it could also be applied to social change! I'm sure some social scientist has applied some mathematical model to quantify social movements, disruption, economics, epidemiology (disease). Your body stores potential energy in fat cells for later consumption, and uses up energy in the form of burning calories, metabolizing food, sweating, giving off heat (measured in BTUs) etc, all the chemical processes required to keep you alive.

So energy is a representation of change in a given context using a set of variables and values.