What he found and published in his 1905 STOR is that space dS and time dT adjust to keep C at C no matter what. Conceptually here's how.
Time aboard that train slows down from dT to dt < dT as the train speeds up to v. But C is C is C no matter what; so we can write C dt = ds < dS = C dT because the speed of light is the same on the train and off the train where dT is the rate of time as it's not moving (i.e., v = 0). ds and dS are the distances the light travels through the spatial dimensions at C speed. Now here's the cool part.
From the fact that C is C on both platforms, inside the train and out, we see that ds/dt = C = dS/dT because both space and time adjust to ds and dt from dS and dT as v increases to keep C at C, inside and out. And that's why that little photon inside the train keeps trugging along at C no matter how fast the train is going. That is, the time (dt) and length (ds) of the train are changing to keep it that way.
It can be shown that W = (C + V)/(1 + CV/C^2) is the relativistic relative speed relationship. As you can see it starts with the classical W = C + V, but the denominator, the relativistic adjustment in time and space, modifies the answer according to relativity.
EXAMPLE: Assume your train is going V = .00001C (a very fast one). So W = (C + .00001C)/(1 + C(.00001C)/C^2 = 1.00001C/1.00001 = C. Ta da. And there you are, the speed of light remains at the speed of light even on that very fast train of yours.