The speed of light is the reference. The spaceship is moving closer to the speed of light, the Earth (compared to the spaceship) less close to it.
Please visit the links from my previous message: if you were on one photon having a speed of light c and towards you arrives another photon with the speed of light c, the resulting speed of the incoming photon you’d see would be just and only c. Because c is the limit. The formulas are behind the last link of the previous post.
As I’ve said, it’s against our intuition, that’s why it’s so interesting. But be sure, you’ll never reach any speed even close to the photon: you’re too heavy.
The question “why” is not answerable in the terms of our everyday experiences (if you were a muon, it would be intuitively “natural” to you, you’d ask instead “why the humans don’t see it”) but we know “how”: see the link I’ve given about the history of Lorentz transformation: the different effects were measured, then the formulas were made that not only match what we measure but with which we can predict the new measurements too.
For example, without using these formulas, one derived more than 100 years ago, GPS wouldn’t work. Not only the “slowing down because of the speed” had to be included, but also “getting faster because of being more away from the Earth in its gravitational field”. The later effect (less gravitation — faster clock) was observed in 2010 on the Earth for the height difference of only 33 cm. To achieve that, the lab “clocks” were among the most precise ever made. But for GPS it’s much more obvious: if you have a 1 GHz processor in your phone (and you probably have something close to that), it ticks 1 billion times per second (one tick per one nanosecond) when it processes in full speed. The GPS clocks keep their tick speed very constant, and they need even higher stability than 1 GHz (their ticks, of course, deviate much less than those in the phones) but imagine just 1 GHz ticks: The difference in the number of ticks between the clock on the Earth and the one on the GPS satellite would be around -7000 ticks per day because it moves faster, and around +43000 ticks per day because it’s more distant from the Earth, which results in +38000 ticks per day more on the satellite. This is equal to 11 km difference in the distance traveled by the signal per day (because the radio waves travel cca 0.3 m during one 1GHz tick, that’s the speed of light).