I think in some sense this is the wrong question to ask. Your watch isn't running slower. The point is that if you're moving, you must be moving relative to something else. Their watch will appear slower to you, and your watch will appear slower to them. The counterintuitive part is that you're both correct.

The reason all this comes about is that both observers measure the speed of light as traveling at the same speed, but they can't agree on the path that the light has taken.

If we use this idea there are some space-time atoms, to explain entanglement. That same concept could explain space-time itself better.

It is similar to how Einstein used anomalies in the application of Newtonian physics to light to postulate a more accurate model. Explaining entanglement might be the gateway to explaining space-time more accurately.