What mechanism will force mechanical watch to tick slower when go fast, due to relativistic effects? To make mechanical watch tick slower, watch tick rate must be changed, oscialtion of balance wheel must be SOMEHOW changed, how would speed  change oscialtion of balance wheel, due to relativistic effects?
I dont understand mechanism between speed and parts inside mechanical watch that will somehow mysteriously start ticking slower?
This video show how watch works.
 A: It is a very common mistake to assume that moving watches slow down, one that is no doubt due to the ambiguous phrase 'moving clocks run slow.' What time dilation actually means is that the time interval between two events occurring in the same place in one frame is shorter than the time interval between the same two events in any other moving frame. Let me repeat that  the time interval is shorter . So a time interval between two events that occur in the same place might be 5s, say, in their rest frame, while in another frame it will be longer, say 6s, depending on the relative speeds of the two frames. So you can see that it is the actual time that differs between the frames. An accurate watch in the first frame will measure 5s not because it is running slow but because the duration is 5s, while accurate watches in the second frame will show that the duration is 6s not because they are running faster but because the duration is longer.
A: As a supplement to Marco Ocram's excellent answer: we are all moving not only in space, but also in time. We have no choice about that: even if we think we are "at rest" in space, we'll be moving forward through time. But different observers may be moving in different directions in spacetime. If we assign $(x, t)$ coordinates to the path of a watch, with the beginning of the path at $(0,0)$, then after one of our seconds a stationary watch will have coordinates $(0, 1)$ whereas a moving watch will have coordinates $(v, 1)$. The vectors $(v, 1)$ and $(0, 1)$ clearly point in different directions and have different lengths. If you mark out 1 unit intervals along the $(v, 1)$ (moving) line, they won't have the same time or space coordinates as they would along the $(0, 1)$ line.
The only complication to all of this is that time and space are not the same thing, and so in practice the "distance" is calculated with $x^2 - t^2$ rather than $x^2 + t^2$. Time comes into it with a "negative" sign (actually the choice of signs for time and space are arbitrary, but they have to be opposite).
This also explains why when you bring the moving watch back it will show a shorter time. The "moving" watch goes around two sides of a triangle, where the "resting" moves along the third side (only in time, not in space). The moving watch travels a longer spatial distance. Time and space have opposite signs, so this corresponds to a shorter temporal distance, i.e. the watch that moved will show less elapsed time.
A: The tricky thing is: it is not the watch that ticks differently, it is time itself.
Let us consider the situation you proposed on the comments: you are on a fast rocket and there is a clock on Earth. What do you see? You see your watch ticking just as usual, while the clock on Earth (which you are looking at e.g. with a telescope) is ticking slower. However, if I am on Earth, I'll see the clock ticking as usual, while I'll see your watch ticking slower.
But not only we'll see the clocks and watches ticking slowly. We'll see everything happening slowly. I'll see you moving slowly, I'll see things dropping to the ground slowly, you'll see me moving slowly, everything slows down.
It is not the watch that changes. It is time itself. Time is not something absolute that governs everything and everyone. It is relative. Time depends on where you are and what you are measuring.
I have written some other posts on this problems. You might be interested in this, this, and this.
A: Your video shows - very nicely - the balance assembly swinging to and fro due to the balance spring. It has a mass (actually a moment of inertia - look at those weights round the rim) that determines how fast this happens.
Suppose I set it up in a laboratory swinging to and fro once every second.
You observe this while travelling past me in a fast train (or plane, or space-ship). Because you are moving you will see the mass of the balance wheel increased by a $\gamma$ factor. The spring properties are the same (under certain assumptions...) so - according to you - the balance assembly takes longer to accelerate through its cycles and the watch ticks more slowly.
