The electric and magnetic field which can support each other to form an entity and propagate in space is called the electromagnetic wave. No one before Maxwell predicted the existence of electromagnetic wave. The remarkable discovery the Maxwell's equations have is the fact that a changing electric field produces a changing magnetic field, and the changing magnetic field can produce the changing electric field.
You may have noticed from the discovery of Maxwell that these two fields can support each other to form an entity (one field producing another field). And such a combination of two fields which can propagate in space carrying energy is electromagnetic wave. Isn't it fascinating? It can propagate in space where there is no medium such as outer space, both fields can coexist and travel. That's how sun's radiation reaches us, it wouldn't have been possible if it required medium to travel.
In Figure 1 below, you can see an antenna (two rods) connected to an ac source. The ac source produces sinusoidal current in the rods, that is in one half cycle one rod is positively charged and another rod is negatively charge. Let's focus on one half cycle only and assume that the upper rod is positively charged and lower negatively charged (the case shown in Figure 1).
The electric field lines originate from the positive charge and end on the negative charge. Since the current is sinusoidal, the charges oscillate in the antenna. That in turn creates the changing electric field and this changing electric field further creates the changing magnetic field. You know that an electric current produces magnetic field encircling it.
Note that the changing electric field is the same thing as current, and the magnetic field encircles it as shown in Figure 1 by the dots and crosses (dots mean field is outward and crosses mean field is into the screen). We know from Maxwell's displacement current that the displacement current is the real current with actual magnetic field surrounding it (and the displacement current is none other than the changing electric field).
The sinusoidal current creates the oscillating charges in the antenna, it's the same as both charges of an electric dipole oscillate. As a result the varying electric field is generated which in turn generates the varying magnetic field and the whole combination of varying electric and magnetic field travels in space or radiated in space.
It is similar to the case we moved a stretched string up and down in mechanical wave so that the generated pulses can travel along the string. Here the charges are accelerated in a direction transverse to the direction of propagation of the wave similar to we accelerated one end of a stretched string whose other end is fixed at support.
A moving charge has both electric and magnetic fields. As the charge moves with constant velocity the electric field at any point changes with respect to space and time, and that's what generates the magnetic field (let's not go to the discussion of the geometry of the magnetic field as that is more complicated) but the fields are not radiated away from the location of the charge because the charge is not accelerated.
While the charge was stationary, there was only the electric field emanating from the location of the charge in the surrounding space. As the charge comes into motion, the charge has the magnetic field now. The charge coming into motion with speed much lesser than the speed of light has no effect on the energy stored in the electric field. So the additional energy in the magnetic field must come from the work initially done to bring the charge into the motion.
