The electromagnetic field is a fundamental quantity, which we cannot explain by simpler assumptions. They are just there and are described very well by Maxwell's equations. Their physical meaning is empircally defined by their influence of charged test particles, and electric charge is a fundamental property of matter too. So it is just there and cannot be explained by something more fundamental or simpler.
In a given inertial reference frame the electromagnetic field can be decomposed in electric and magnetic components, which are described as usual three-dimensional Euclidean vector fields ##\vec{E}## and ##\vec{B}##. For a region in space without charges and currents, Maxwell's equations have solutions, describing electromagnetic waves like
$$\vec{E}=\vec{E}_0 \cos(\omega t-\vec{k} \cdot \vec{x}), \quad \vec{B}=\vec{B}_0 \cos(\omega t - \vec{k} \cdot \vec{x}),$$
where the relation between the frequency and the wave vectors is
$$\omega = c |\vec{k}|,$$
which means that these plane waves move with the universal constant ##c## appearing in the Maxwell equations (in a proper choice of units like Gaussian or Heaviside-Lorentz units; in SI units this is a bit hidden in the conversion constant between SI units and more natural units, but it turns out that the only physically relevant quantity is again a universal constant with the dimension of a speed, ##c=1/\sqrt{\mu_0 \epsilon_0}##). Because this speed is the phase velocity of the plane electromagnetic wave in a vacuum, it's usually called the speed of light in a vacuum, although relativity teaches us that it has a far more general meaning as a conversion factor between time and distance units. In even more natural coordinates one would measure distances in terms of the travel time of light signals (and in the SI one does so in fact by defining ##c## to a certain value matching as precisely as possible the older definition of the time units second and distance unit metre).
Finally the Maxwell equations also tell us that we must have
$$\vec{k} \cdot \vec{E}_0=\vec{k} \cdot \vec{B}_0=0, \quad \vec{E}_0 \cdot \vec{B}_0=0.$$
One gets even more precisely
$$\vec{E}_0 \times \vec{B}_0 \propto \vec{k}.$$
So the electromagnetic waves are transverse waves, and what's oscillating is the electromagnetic field (or in the fixed reference frame, we've used above to describe this particular solution of the Maxwell equations, the electric and magnetic field components ##\vec{E}## and ##\vec{B}##).
