I gave this explanation on another website to some laymen and they liked it, so I'm going to copy the gist of it here. As usual, corrections to my layman's explanations are welcome.
What happens when a clock's or mechanical watch's hand ticks one second is that something called the balance wheel inside the clock, keeps very accurate time. Overview here. The mechanical wrist watch in general is a rather remarkable bit of engineering that keeps very close to precise time. It requires energy to keep running, either stored mechanical energy by winding, stored in the mainspring, or stored by battery.
But if we take a closer look at what happens when a balance wheel swings, that is, we look really close, it all goes down to the quantum level. Protons and Electrons send signals to each other by photons. That's how the tension on a spring is stored. This can get a little complicated, so I'll just refer you to this question here.
Now, velocity is relative. The watch traveling at .5c thinks it's standing still and thinks you're the one moving, but if we use your frame of reference and we say the watch is moving and look at everything from your perspective, the protons and electrons are sending each other signals and energy by photons, the electromagnetic force carrying particle. And when the watch is "moving" at .5c, depending on the placement, of the electron (and the electron is also everywhere at once, but lets not go there), but say it's ahead, in the direction the watch is moving. The photon leaves the proton at c, cause photons always travel at c, but it has to catch up to the electron that's moving away from it at 1/2c. That takes twice as much time (from your perspective). Or, say, the electron is on the other side moving towards the spot where the proton is emitted the photon it takes 1/2 the time. When averaged out over the sphere around the proton, the time it takes for protons and electrons to tell each other where they are works out to the Lorenz contraction when the atom is moving (relative to you), so when a mechanical spring moves past you at 1/2c, it takes the protons and electrons 15.5% longer to tell each other where they are and exchange energy and information. That means the spring responds 15.5% slower.
Now, because every proton and every electron that is moving at .5c relative to you is affected in the same way, the person wearing the watch doesn't experience the slowing down because everything slows down equally, or by his perspective, he's standing still and nothing is slowing down. That's, at least, the special relativity explanation. I'm not going to try to explain in general relativity cause that doesn't really change your question, but general relativity does explain how both objects can see the other moving slower, even if they meet up again at a later time.
Short answer: Everything's quantum. Drop a cinder-block on your foot. That's a quantum experience. It can all be modeled by fields and quantum forces interacting. It's just quite a bit easier to just look at it in the mechanical and stick with Newton's explanation.