Think about this
In your brain there is a clock too. In other words, everything has a clock in itself. The clock in this meaning is the electron and molecule activity. Both gear and battery is electron and molecule activity. And inside your brain is another kind of electron and molecule activity
Time dilation is about the reality that all electron and molecule activity of everything will decrease speed when that thing move. So when you move with your watch. Your brain activity will slower in the same rate as your watch. And you will not notice that your watch are slower
But if someone stay still and observe your watch. Electron and molecule activity of that person is in constant rate. So they will notice that your watch is slower than normal
The cause is speed of light. Electron and molecule activity take place around atom. Electron move between atom cannot move faster than speed of light. So when you move. The speed of electron that jump between atom need to be slower to keep its raw speed under speed of light
And about twin paradox
Imagine that you cell multiply and dead everyday is the biochemical reaction, and it is another kind of electron and molecule activity. So yes, the aging process is slower by time dilation too
Brandon Enright, how's it then when I'm outer-space, and travelling in a rocket, I don't age as much as my twin does?
– Nov 20 '13 at 21:35