What is the exact mechanism by which time dilates? What is the exact mechanism by which time dilates for a fast moving object?
 Can the time dilation be explained by any theory other than relativity?
 A: Your question is a natural one to ask, but it has no answer. It is a bit like asking by what mechanism the angles of a triangle always wind up adding to 180 degrees (in Euclidean geometry). There is no mechanism for that - no one is going around checking all the triangles to make sure their angles add up right. It is just a logical consequence of the theory of geometry. It can be proven, and you can understand it intuitively, but there's nothing actively making it happen. There is similarly no mechanism behind time dilation. 
In the very early days of relativity, before Einstein, people had come up with the idea of length contraction, and they had a mechanism for that having to do with the ether pushing on the atoms or something. It turned out that was all wrong. There is nothing pushing the atoms to make them contract. Lengths just look different in different frames. Similarly times look different in different frames. 
This is not too different from the way that if you tilt a rectangle by 90 degrees its length becomes its width and its width becomes its length. By what mechanism do they swap? None. That is just how geometry works. By what mechanism do times and lengths change? None, that is just how Minkowski space (the geometry of special relativity) works.
The primitive concept to learn is called the spacetime interval, which is like a distance and has its own version of the Pythagorean theorem to describe it. Spacetime is really about events with spacetime intervals between them. These intervals are neither length nor time, but instead a combination of in the form of $\Delta s^2 = \Delta x^2 - c^2\Delta t^2$. This is the thing that's always the same, but in order for it to be the same in all frames, space and time need to switch around.
Suppose, for example, that I snap my fingers, wait a moment, then snap them again. To me, these two things happened in the same place, so $\Delta x = 0$. However, if I am in a car and drift past you, you think they happened in different places, so $\Delta x' = v \Delta t'$ where $v$ is the speed of the car. The primes ($'$) are there to indicate these are your measurements. The basic law of relativity is
$$\Delta x^2 - c^2\Delta t^2 = \Delta x'^2 - c^2\Delta t'^2$$
and plugging in what we already know about the $\Delta x$ values, we get
$$0 - c^2\Delta t^2 = v^2\Delta t'^2 - c^2\Delta t'^2$$
or
$$\Delta t = \Delta t' \sqrt{1 - \left(\frac{v}{c}\right)^2}$$
this is time dilation. I measure a shorter time between the events than you do just because we both measure the same spacetime interval. And because I measure a shorter time, you think that my clocks are going slow. Of course nothing is happening to the clocks - they work fine. It is just a fact that the interval $\Delta s^2$ is the same for us and so times are different. There's no mechanism behind us measuring the same interval, either. That's just a postulate of relativity the way, for example, the parallel postulate is a part of Euclidean geometry.
If you continue on in this manner, assuming the interval is invariant, you can calculate all the other formulas people use in introducing relativity, including length contraction, relativity of simultaneity, velocity addition, Lorentz transformations, and whatever else you want. They are all just pieces of the same idea.
There is a lovely laymen's book called General Relativity From A to B by Robert Geroch which describes the geometry of relativity at length and with pictures. I recommend it highly as a starting point for relativity.
In answer to your last part of the question, I suppose other theories could explain time dilation, but relativity is simple, beautiful, time-tested, has made accurate predictions and passed stringent experimental tests, and is deeply embedded in all of theoretical physics. It is correct at all scales we've tested, and any deviations need to be hidden at very short lengths, very high energies, etc.
A: To add to Mark Eichenlaub's nice answer...
Suppose that in the Euclidean plane, you have two people, $P$ and $P'$, located at the same point (or as near as possible), but they're facing in different directions. Each of them imagines the usual Cartesian coordinate axes, say with $x$-axis to the direction they're facing, which we'll call depth.
Suppose also that $P'$ has a rod held outward:

$P'$ considers the $x'$ axis to be depth, naturally says to $P$ that the rod has a depth $L$. But $P$ disagrees! Clearly the rod has a depth of $L\cos\beta$--that's how far as it goes along the $x$-axis, after all!
Now let's say $P$ has his or her own rod along the $x$-axis, and measures its depth to be $M$. Now $P'$ will disagree--according to $P'$ the rod now has a depth $M\cos\beta$. Each claims the other's measurements are wrong by the same factor.
What is the mechanism of depth contraction?
Now it seems almost silly to ask: the cause is that they're facing in different directions and and insisting that the direction they happen to be facing is the correct depth. There isn't anything more mysterious then them facing in different directions, with their coordinates related by a rotation:
$$\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}
\cos\beta & -\sin\beta \\ \sin\beta & \cos\beta \\ \end{bmatrix}
\begin{bmatrix}x\\y\end{bmatrix}$$

Here's the thing: in spacetime, the "direction you're facing" is a velocity. For example, one can use a vector $[t;x] = [5\,\text{s};10\,\text{km}]$ to represent a velocity in the $x$-direction of $10\,\text{km} / 5\,\text{s} = 2\,\text{km/s}$. This part isn't even special-relativistic; it's just as valid in Galilean (Newtonian) spacetime.
The part that is special-relativistic is that STR says that spacetime rotations (more commonly called boosts) work hyperbolically:
$$\begin{bmatrix}ct'\\x'\end{bmatrix} = \begin{bmatrix}
\cosh\alpha & -\sinh\alpha \\ -\sinh\alpha & \cosh\alpha \\ \end{bmatrix}
\begin{bmatrix}ct\\x\end{bmatrix}$$
The difference coming from the different way distances work in Euclidean space vs. special-relativistic spacetime
$$\begin{eqnarray*}
\text{Euclidean (Pythagoras)}\quad &\Delta s^2 = \Delta x^2 + \Delta y^2&\quad\text{cf. }\cos^2\theta + \sin^2\theta = 1\\
\text{STR (Lorentz)}\quad &\Delta s^2 = -c^2\Delta t^2 + \Delta x^2&\quad\text{cf. }\cosh^2\alpha - \sinh^2\alpha = 1\end{eqnarray*}$$
A Lorentz transformation is a particular case of a rotation in spacetime. Usually, it's written in different form, but this is equivalent: $v = c\tanh\alpha$ gives $\gamma = \cosh\alpha$ and $\gamma v = c\sinh\alpha$.
The cause of time dilation is then simply that observers are 'facing' different directions in spacetime--and again, direction in spacetime is a velocity.
A: 
What is the exact mechanism by which time dilates for a fast moving object?

There's no intuitive explanation nor any other special mechanism to describe SR effects other than the unification of spacetime. The view is completely based on its postulates. There's no absoluteness in time. There's no preferred reference frame meaning that every observer agrees with his very own clock which he carries. $c$ is the fastest possible way of information by which two observers can communicate with each other. Basically, this effect can be explained starting with an event and drawing spacetime diagrams between inertially-moving observers on how they observe the light from that event.
Say for example, a bomb blasts between an observer at rest and and observer moving at constant velocity relative to the stationary observer. Both the observers agree that the event happened. But, they simply disagree with "when the event occurred?" and this is an intro-version for this question...

Can the time dilation be explained by any theory other than relativity?

There haven't been a new theory for showing just the time dilation. Because, it wouldn't be a theory. A new theory is something that explains every prediction of its predecessor and the unexplained of the old theory. But, there were many experimental verifications for time dilation like the extra lifetime (i.e) late decay of muons and also a comic, but encouraging one: slowing down time when you travel in car (though I doubt the radical change in the time interval, it's still good for starters to believe)

As for me, this is a question which requires a very large answer. Here are some suggestions which may help you understand the mechanics:
Help Me Gain an Intuitive Understanding of Lorentz Contraction
What are the mechanics by which Time Dilation and Length Contraction occur?
Time Dilation - How does it know which Frame of Reference to age slower?
A: Special relativity itself makes it clear that absolutes cannot be detected, such as being at absolute rest in space, or the inability to detect absolute motion. This therefore prevents one from having an absolute understanding of special relativity, since that which special relativity reveals does not extend to the point of absolutes. Thus the absolute foundation of which special relativity resides within, is not being exposed. 
Therefore the true cause of time dilation, is also not being exposed. 
However this problem can be resolved. We must start at ground zero. We must start with absolutes. 
Picture an absolute 4 dimensional space-time environment. Then picture absolute motion ongoing within that environment. Let it be said that this ongoing motion is common to all objects, and that the magnitude of this ongoing motion is equivalent to the magnitude of motion of which light has as it moves across space. This is known as (c). Despite all objects being constantly on the move within space-time, change in their individual directions of travel, is still an open option.
If one then proceeds to examine the outcome of such circumstances, the outcome is the phenomena known as "Special Relativity". Thus an "absolute" foundation, leads to a "relativistic" outcome.
Change in direction of travel, determines the balance between motion across space, and motion across time. Change in direction of travel, also results in rotation within space-time. The rest is mere details.
See https://www.youtube.com/embed/KKAwpEetJ-Q?list=PL3zkZRUI2IyBFAowlUivFbeBh-Mq7HdoQ for the 1 hour 37 min coverage of the analysis of the outcome of the absolute foundation.
A: Our visible universe is made up of 3 dimensions of space and 1 dimension of time. All objects in the universe exist in this space-time continuum. Space and time are not 2 separate entities irrelative of each other. Space and time are intimately united as one entity called spacetime. As mentioned earlier, all stationery objects experience their time in this visible universe. But no object in the universe is stationery. Each and every object is moving with respect to one another. When an object moves at a speed, it experiences time slowly. This is because apart from the visible universe there is an invisible universe intertwined within the visible universe. Even that invisible universe is made up of space and time but are combined differently and thus having different nature.
    In the visible universe (VU), space ≠ 0 and time ≠ 0. Space comprises of 3 dimensions length, width and height. So, in total there are 4 dimensions.
    But in the invisible universe, space ≠ 0, but time = 0 and even in this universe there are 4 dimensions. But these dimensions are different from the dimensions in the visible universe because space and time are tied up in different ways. In the invisible universe, no events can happen since time = 0, but matter can exist since space ≠ 0.
    The slowing of time of a moving object is due to the fact that a moving object does not fully spend its time in the visible universe, it spends some of its time in the invisible universe. It is not that things happen in a slow motion for a fast moving object, it is that it does not exist within the visible universe for some definite amount of time.
http://www.scribd.com/doc/138077520/Space-Time-2013
