The reason why we need special relativity is, that wave equations are not invariant under galilean transformations. The reason for this is, that the veolcity of a wave is determined by the medium. That means that an observer with a relative speed to the medium v>0, simply observes the wave going at the wrong speed. So he cannot describe the wave correctly. Now, with a medium, one can always get away with saying that the medium is the relevant reference frame and that therefore we don't have to worry about galilean transformations.
With light, things are different, since light doesn't require a medium. Hence, either the wave equation of light is wrong or the galilean transformation.
Since the partial derivatives are coefficients of a dual vector, the transformation that will leave this equation invariant, will be orthogonal with the metric of special relativity (due to the relative minus sign between the time and space components).
One can now simply generalize the galilean transformation, by putting variables in front of each term, and then use the chain rule of the partial derivatives to work out the Lorentz transformation.
Only the wave equation with velocity c is then invariant under the so found Lorentz transformation. All wave equation with speeds different to c are still not invariant. This means that the speed of light has to be the same for all observers, or otherwise we still wouldn't have an invariant description of light.
I hope this helps.