Let me try to explain it differently.
First of all the "speed of light" refers to the phase of a electromagnetic wave. A plane wave propagating in ##z## direction is described by (in Heaviside-Lorentz units)
$$\vec{E}(t,\vec{x})=\vec{E}_0 \cos(\omega t-k z), \quad \vec{B}(t,\vec{x})=\vec{e}_3 \times \vec{E}_0 \cos(\omega t-k z).$$
Maxwell's equations tell you that
$$\omega = c k$$
and this implies that the planes of constant phase, defined by
$$\omega t-k z=\text{const} \; \Rightarrow \; v_{\text{phase}}=\mathrm{d} z/\mathrm{d} t= \frac{\omega}{k}=c.$$
Now this plane wave must come from somewhere. Indeed it's a solution of the Maxwell equations describing an electromagnetic wave emitted from some very far source.
Now the remarkable thing with the Maxwell equations is that they contain as a parameter this phase-speed of electromagnetic waves (in vacuum), ##c##, which is considered a fundamental constant of nature.
At the same time from classical mechanics we expect that all the physical laws must look the same in any inertial frame of reference. Now suppose you observe the far-distant light source when moving with constant speed ##v## along the ##z## axis towards this light source. From naive Newtonian reasoning you'd argue that now the phase velocity must be ##v_{\text{phase}}'=c+v##, but that contradicts the invariance of the Maxwell equations when transforming from the coordinates as measured in the old frame to the coordinates as measured in the frame, where you are moving.
This was the problem physicists where struggling with since Maxwell has written down his equations, and the final solution was Einstein's famous paper about special relativity: We have to change the description of space and time in such a way that the Maxwell equations are unchanged when transforming from one inertial frame to another one, and thus the phase speed ##v_{\text{phase}}'## must be the same as in the original frame ##v_{\text{phase}}'=v_{\text{phase}}=c##.
In other words the phase speed of electromagnetic waves as measured from any inertial observer is independent of the velocity of the source of this electromagnetic waves.
This is achieved by the Lorentz transformations for time and spatial coordinates described in #7. It turns out that also ##\omega/c## and ##k## must transform as time and space coordinates, i.e., in our setup where the observer in the new frame moves towards the source
$$\omega' = \gamma (\omega + \beta c k), \quad k'=\gamma (k+\beta \omega/c).$$
Since now ##\omega=c k##, ##\beta=v/c## and ##\gamma=1/\sqrt{1-\beta^2}## you get
$$\omega'=\omega \frac{1+\beta}{\sqrt{1-\beta^2}} = \omega \sqrt{\frac{1+\beta}{1-\beta}}, \quad k'=k \sqrt{\frac{1+\beta}{1-\beta}}.$$
You thus get also for the observer moving towards the source
$$\omega'=c k',$$
but ##\omega'>\omega##. This means tha phase speed in the new inertial frame is indeed the same as in the original frame, i.e., ##v_{\text{phase}}'=\omega'/k'=c## but the frequency is larger than in the old frame. This is the Doppler effect for light: Moving towards the source shifts the light somewhat towards higher frequencies (i.e., it's "blue-shifted", because light that occurs blue to us is an electromagnetic wave with higher frequencies than, e.g., red light). In the same way you can show that for an observer moving away from the source the light gets red-shifted (here simply ##v## is replaced by ##-v## in the Lorentz transformation).
So the phase velocity of em. waves stays the same when changing from one inertial frame to another one, i.e., it's the same for any observer no matter, which speed the light source has in his rest frame, but the frequency gets shifted according to the Doppler effect for em. waves. This is in accordance with Maxwell's equations, which should not change by changing from one inertial frame to another moving with constant speed against the first frame, and thus the phase speed of light, ##c##, must be the same in all frames. This is achieved by using the Lorentz transformations for changing the time and space coordinates when transforming from one inertial frame to the other rather than the Galilei transformation valid in Newtonian mechanics.