We really don't know. The nineteenth century physicist James Clerk Maxwell tried to find some sort of mechanical explanation that tied the motion of electrons to magnetic fields but he was unsuccessful. Today most physicists would say that the rules of electromagnetism are one of the fundamental facts about the world that we can't explain, but we can use to explain many other things. For example, everything we know about light is based on our understanding of the relationship between electric and magnetic fields.

Here is Richard Feynman --- one of the greatest physicists of the twentieth century --- answering a question very similar to yours. I hope you find this as interesting and inspiring as I do:

Actually there are two ways to generate the magnetic field: magnetic material and electric current. The magnetic materials produce the magnetic field because of their intrinsic magnetic moments. The other way you mentioned is the electric current. Before I talk about the electric current, it is worth mentioning electromagnetism. As a matter of fact, the electric field and magnetic field are tightly connected to each other. A moving electric field (electric current) can produce the magnetic field. A changing magnetic field can also produce the electric field.

Any static charge (such as an electron) can produce the static electric field. A moving charge (electric current) can produce the magnetic field which will affect the motion of the moving charge in return. These two fields are tightly connect to each other.

Electricity and magnetism are a single force. A changing or transforming electrical field creates a magnetic field, and vice-versa. You can derive the equations of magnetism from the equations that describe electrical forces and Einstein's equations of special relativity. Electrical currents, because they contain moving electric charges, represent time-variant electric fields, which in turn create magnetic fields.