Can someone conceptually explain time dilation? Can someone intuitively explain how physically time dilation happens?
Please don't explain about the invariant speed of light and the mathematical background, I am familiar with that. I just can't imagine how this time dilation process is happening physically, and I can't understand how to distort my mind to understand it!!!
Sorry for this question, it probably sounds like, "Why there are positive and negative charges", but I have to ask!
What could make the question clear is the equivalent one: how could be that the two moving observers do not agree on the simultaneity of events and one sees the other in slow motion but he also knows that the other see him in slow motion. How to imagine the reason of this time dilation, probably something similar to banding of space leading to contracting lengths but for the time. It would have been more clear if the time difference was unsymmetrical and if observer A sees me in slow motion then I should see him in faster motion, so ok I agree that my time has being delayed and he agree that too, but this is not the case.
 A: One result of special relativity is that the magnitude of all 4-velocity vectors $\vec{u}$ is the speed of light.  Written with the (-,+,+,+) signature:
$$\vec{u}\cdot\vec{u} = -c^2$$
One way to think of this is that everything is always moving the speed of light in some direction.
When I stand still, I move the speed of light in the time direction.  My clock advances as fast as possible.  When I look at other observers in other moving frames their clocks all advance more slowly than mine.
Imagine someone moving close to the speed of light.  Their clock seems to hardly advance at all relative to mine.
If I start running, my 4-velocity vector is still the speed of light long.  Now it has some non-zero component pointing in the space direction to account for my motion.  That means the time component of my 4-velocity must have shrunk.  My clock does not advance as fast as it used to, relative to everyone else, who remains standing still.
A: I think it might be more instrctuive for you to show yourself what is going on. If you work through the following example you'll come out with a decent understanding of the phenomenon.
Consider a light clock onboard a ship that is moving at velocity v relative to an observer. Te light clock works by boucing light vertically between two mirrors spaced one light second appart. Given the postulate upon wich all of special relative is based "the speed of light is the same in every inertial frame" You can work out that the stationary observer sees the light in the clock to travel a greater distance tan the one moving with the clock. Therefore because the speed of light has to be the same in both reference frames time must be slower in the movig frame. i.e.
distance = speed x time
distance is less in te moving frame, speed is the same so less time must have passed. Therefore time is moving slower in the moving frame, hence time dialation. If you do the maths for the example you will be able to derive an equation for it.
A: Clocks don't measure time and tape measures don't measure distance. They both measure the metric along a curve. Just like how a tape measure depends on the path so does a clock.
Take 4d spacetime, look at the path of the clock in 4d and note that at some of the events, it ticks. So at one point the clock ticks. And how far along the path should the clock go before it ticks again is 100% the entire question.
And it is based on two things. How deep a gravity well it is in, and how much of the curve is there between the two events, the start point and the end point of the curve in 4d.
Now curves are longest when going in straight lines, that's just how geometry dictates lengths in a Lorentzian geometry. So you can break down the curve into pieces and replace each little piece with a straight line and you've over estimated the length (in Euclidean geometry you'd be underestimating the length). So now you just need the length of each piece. You get that from the metric. Why? You can imagine there is some real time and the clocks are mean and just don't tick that way. Instead they tick based on a metric. The metric literally tells them how to tick. They tick because they measure the metric rather than measuring time.
You can imagine that the clock has to go along a 4d path and that the space and time flow through it as it ends up at different events. And for each little bit it computes the metric of that little bit and keeps a running total and when it gets to a certain total it ticks.
All you have to do is accept that clocks literally measure $\mathrm ds=\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}$ for each little piece and add it to a running total and then tick when that running total crosses the cut off running total.
And everything else too. When something is supposed to happen at a certain rate instead of waiting a certain amount of time, you go along a curve in 4d spacetime, compute $\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}$ along the curve and instead of doing it y times a unit of time you do y times every unit of $\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}.$
A: Forget everything you know about relativity for a moment. Assume that there is some absolute frame of reference in the universe. That the subatomic particles and waves we are made of are themselves made out of pieces on a 3D chessboard.
Things that keep "absolutely" still stay on the same squares (well, cubes). Things that move will jump from square to square along the chessboard.
This begs the question: does light travel at a constant speed with relation to the chessboard (like sound does, relative to the air it travels in), or with relation to the source that generated it (something like ejected particles)?
The latter can be easily shown to be false by timing how long it takes light to reach you given a moving light source. What about the former? Well, on a fast-moving spaceship, you would expect light to travel much faster going towards the rear of the ship than towards the front. But this isn't the case. No matter how fast you are travelling, light seems to move the same speed no matter who creates it or which direction it is sent in.
How can this be? Most people believe there is no "3D chessboard" and that it is simply a physical law that the speed of light in all directions be the same for all observers. Personally, I believe that the "3D chessboard" does exist in some form (something that admits curvature), but we can't observe any differences in the speed of light, because our clocks and rulers are implicitly calibrated by the very same speed of light! It is like trying to measure inflation by seeing how many \$10 notes it takes to buy a \$50 note.
I will present an oversimplified picture of time dilation to make my point. The physical processes that make a clock tick depend on the speed of light. If light travelled slower, the clock would also tick more slowly, so it would be undetectable! Actually the reality is more complicated, because light would not be uniformly slower but rather slower in certain directions and faster in others, so lengths in various directions get distorted, as well as the time it takes for signals to reach an observer. But everything cancels out in the end, meaning that the observer cannot observe her own speed relative to the "3D chessboard".
If you go down this path it is very important to distinguish between distance and time as measured by us, and "absolute distance" and "absolute time" as measured against the absolute frame of reference. The equations of Relativity deal with distance and time as measured by us. "Absolute distance" and "absolute time" are things we cannot measure and do not have any units for.
Maybe there is no absolute frame of reference (I'm not aware of any evidence supporting it). But even so, I think it's a useful stepping stone to getting your head around Einsteinian Relativity.
A: Pekov: Without going into mathematics (which you may know more than I do), this is an as fundamental question as "how mass curves space/time". Because the effect comes out of that curving. 
One way is, you can accept curving, and then trust the math.
There is no yet known physically describable mechanism of "how the curving actually takes place". So, we are out of luck here.
However, how I soothe myself is this way - Any object in motion feels some kind of stress due to the speed through space. This stress may arise from the fact that the nature (the object, its fields, space around it etc.), has to know changed position of the object at every moment as it moves. To do that, nature has to continuously do some heavy computations. Due to this stress, all events in the moving system slow down a bit depending upon how fast it is moving.
I can not do better than this - a philosophical answer for a philosophical question.
