I know how to get the wave equation of light from Maxwell's equation, but I never understood why it is called the solution of the Maxwell equations. If say we have a positive charge standing still from our frame of reference, it generates an electric field that is solution of the Maxwell equations (Gauss law) but it does not propagate, it remains "fixed". I don't understand the difference between this electric field and the ones that do propagate freely through space, and how can it get "de-attached" from its source?
The field of a charge in otherwise empty space obeys the wave equation everywhere but at the position of the charge. Because it has zero frequency it cannot propagate. It also is a solution of Poisson's equation, which has no propagating solutions.
To generate propagating, non zero frequency solutions, time dependent currents are needed.
Maxwell equations are a set of 4 vectorial differential equations. Given a charge distribution and current (basically think of filling up space with things) that may or may not move, those equations will spit out an electric field and a magnetic field.
If you arrange your charge distribution in some specific ways, you can get propagating fields, or stationary ones. You are talking of a special distribution, which is empty space. Solving maxwell equations there gives us the wave equation that you know.
Now, if you want to have a propagating field emitted by something is space, you have to have that thing moving. Otherwise, the solution will not be time dependent (this is obvious from the maxwell equations themselves).
For example, you can think of an oscillating charge, in that case if you make the amplitude of the oscillation small enough and look far away, you can approximate the electric and magnetic field by a spheric wave with wavelength equal to the amplitude of the oscillation of the charge.