How does an electrical field really work?
There are two formulations that describe the known data on electric and magnetic fields.
a) The classical electromagnetic theory ruled by Maxwell's equations . This works well in describing the macroscopic data, of which the electric field is a component.
b)The quantum mechanical formulation that leads to an explanation of how fields are built up, which is necessary to explain effects like the "photelectric effect", the behavior of atoms and molecules, the internals of atoms and molecules.
For a) the electric field is a fundamental component of the behavior of matter.
For b) the electric field is built up coherently by innumerable virtual particle exchanges, mainly virtual photons, between the generators of the field and the detectors of its existence, so it is not fundamental. Charge is fundamental in this framework, and charge is quantized (+/-1/3, +/-2/3,+/-1)in absolute value electron charge units. That is why it is a quantized theory of the world.
How come that an electron can exert a force on another electron without physical contact?
For a) it is an action at a distance , the field of the electron exerts a force on other charged matter; similar to classical newtonian gravity, where the masses exert a force on each other.
What is it in an electron that creates the field, where does the energy for doing the work come from?
It is the charge of the electron. When we are talking of electrons we are really in the realm of b), quantum mechanics, because its size is of the size where quantum mechanics has to be used to understand the data.
In QM language the electron, when looked at individually, is continually exchanging virtual photons with the boundaries of its containment. Virtual means that energy and momentum are not conserved because nothing real is exchanged with the other electrons/ions except an information "I am here". When many electrons are involved, the surface of a charged metal sphere for example, the collective electric field is built up out of those exchanges.
The energy was supplied in this case by the experimenter who provided work to separate the electrons from the rest of the molecules, turning them into ions.Either by the triboelectric effect or the classical generators of electricity, using magnetic fields and providing a current of electrons in metals. Ultimately it is kinetic energy turned into electric energy. ( actually sun energy stored in fuels or water works, turned into kinetic energy ...)
Now magnetic fields, used to generate most of our electricity, are a bit of a different story, but similar and again needing quantum mechanics to be understood. In scenario a) they also are fundamental.