According to Wikipedia:
"The electromagnetic field is a physical field produced by electrically charged objects. It affects the behaviour of charged objects in the vicinity of the field.
The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature (the others are gravitation, the weak interaction, and the strong interaction).
The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz Force Law."
Think of it this way. If you pick up a pebble, and raise it to a height, you are increasing the potential energy of the pebble by doing some work (lifting the pebble against the gravitational field of the earth). And when you drop it, the potential energy of the pebble is converted to the kinetic energy as a result of the motion of the pebble. If you're dropping the pebble into water, this causes ripples, as a result of transmission of kinetic energy from the pebble to the molecules of water at the point of contact. Those molecules transmit the kinectic energy to surrounding molecules and the ripples propogate.
So, you see, it's all a result of transmission of energy from one form to another. In the case of your example, electrons containing electrostatic energy (analogous to potential energy) are accelerated through an electric field as a result of electromotive force. This causes electrostatic energy to be converted to electromagnetic energy as a result of movement of electrons. The EM energy (kinetic energy of the electron) is merely a exhibition of movement of the electron and nothing else.