Light is an electromagnetic wave, which means that its behaviour follows straight from the rules of electromagnetism. The speed of light is just a property that emerges from the relationship between electric and magnetic fields; there's no free parameters that you can fiddle with to speed it up and slow it down. A light wave moving any faster or slower than $c$ is a light wave that doesn't obey Maxwell's equations.
There are two ways that one could perhaps deal with this. You could keep light behaving like a classical moving object, and have the laws of electromagnetism change depending on your speed (this might be a little troubling; you are held together by electromagnetism). Or you could keep the laws of electromagnetism the same, and instead alter the transformation that happens when you change velocity, having lengths contract and times dilate in just the right way to keep light moving at $c$.
Nature does the second one, and the resultant transformation is the Lorentz transformation you know and love.