Oooh I think I know some of this. I don't care if I don't, I'm answering.

The key is time, right? My relative time is different than the earth's relative time.
Yeah, which also means your perception of space is different to an observer on Earth. As time expands, space contracts. You measure the light travelling at c, which would lead you to think that I would measure it from Earth at c+10km/s, which defies logic. But you measure the distance differently to someone on Earth, a direct consequence of the difference in time. You would measure the distance between Earth and beam differently (shorter) than someone on Earth, and that accounts for the extra 10km/s in apparent speed above c.

Or something like that.