Do physicists understand why time slows down the faster the velocity of an object? Like do they understand the mechanism behind it? Is there a mechanism behind it? (for a lack of a better word) Or is it just thought to be a law without any further explanation?
 A: The slowing down of time in special relativity (and the corresponding Lorenz contraction of lengths) is a consequence of the empirical fact that the speed of light in a vacuum is the same for all inertial observers. Why is the speed of light in a vacuum the same for all inertial observers ? Well, that's just how our universe works. It doesn't have to be like that, but that's just how it is.
A: The deeper reason for it is that the invariant distance between 2 events in spacetime is:
$$ (\Delta s)^2 = (c\Delta t)^2 - (\Delta x)^2$$
What that means is that any observers can measure the time and distance between 2 events, and they will all get the same $\Delta s$, even though they do not agree on spatial separation (obvious), nor on temporal separation (astounding).
The good news is that, while this is very non-intuitive, it actually makes the laws of physics much "nicer" than what most everyone's intuition says, which is that time is a universal parameter that ticks away uniformly for all things in a flat 3 dimensional space.
A: Time dilation is a logical consequence of two postulates:


*

*The laws of physics are the same for all inertial observers. This is equivalent to saying that there is no preferred inertial reference frame in our universe. People used to think that there was 1 unique reference frame in which the laws of physics are formulated. They thought you had to do your calculations in this unique reference frame and then transform to the reference frame you are working in.

*The propagation speed of physical interactions is finite and equal to the speed of light. This is the principle of locality: an object is directly influenced only by its immediate surroundings.
Time dilation seems counterintuitive because in our daily lives everything moves much slower than the speed of light. To us the propagation speed of physical interactions seems to be infinite. If we replace the speed of light with this infinite speed, we obtain Galilean relativity where time is absolute.
If we lived in a universe where the speed of light was much much slower, special relativity and time dilation would be very natural to us all and you would be asking how, for incredibly slow processes, time seems to be the same for everybody.
A: At present we must accept the principles of special relativity as is in order to formulate a theory at all.
I'd like to move to a more general perspective, and then back to special relativity specifically.
About progress in fundamental science:
It seems to me that starting with Isaac Newton fundamental science started rolling, and started picking up momentum (pun intended).
Over the past centuries we have experienced pretty much avalanches and avalanches of progress in fundamental science.
So that is the pattern that we have become accustomed to. For example: why is it that when water freezes it expands a little? All other substances just shrink with decreasing temperature. When fundamental science developed an understanding of how the hydrogen atoms and the oxygen atom form a water molecule, and an understanding of the interactions of water molecules with neighbouring water molecules, there and then it was understood why water expands when it freezes.
We have become accustomed to a pattern of fundamental science finding ever deeper insights.
Relativistic physics was introduced more than a 100 years ago. You kind of expect fundamental science to have moved on to an even deeper level.
My assumption is that that is the background of your question. We can explain so much, can we explain the Minkowski metric too?
For comparison:
Kepler's laws of celestial mechanics. Kepler published his three laws of celestial mechanics over a period from 1609 to 1619. It may wel be that at the time there were scholars who thought of Kepler's laws as laws that must be accepted as is. You might like to try and find a deeper level, but if there actually isn't any deeper level you are wasting your time.
Decades later there were scholars who had made some progress. Scholars such Christopher Wren, Edmund Halley, Robert Hooke were aware that Kepler's third law strongly suggests that gravity is an inverse square law. But to show that an inverse square law of gravity gives rise to ellipse-shaped orbits (as described by Kepler's first law) you need differential calculus, and at the time differential calculus wasn't developed yet. Wren, Halley and Hooke had a good hunch; they had a good clue to follow.
As we know now, Isaac Newton had already moved all of mechanics to a whole new level. Upon publication of the Principia these insights became available for all scholars of the time.
Here is why I have spent so many words on Kepler's laws.
Looking at Kepler's three laws it is not immediately apparent that a deeper level of understanding is possible. But of course there is a deeper level.
Special Relativity
Is it concievable that at some point in the future a deeper insight is reached, such that the Minkowski metric is explained in terms of that deeper theory? 
I think that is concievable; you don't know what you don't know. 
It's definitely wrong for anyone to make a blanket statement along the lines of "There is no mechanism, it is the structure of spacetime." That's like a contemporary of Kepler stating: Kepler's laws are the final laws, those laws are the way the universe is."
Then again, if there is a deeper level, we don't have any hunch how/where to look for it. So in terms of prospects of finding a deeper theory, it's not looking good.
A: Some things you might like to know in relation to this are as follows:
The starting point for all of special relativity is the assumption that the speed of light seems to be the same for everybody, regardless of what speed they are moving relative to each other. From that assumption, with just some basic maths- nothing more difficult than Pythagoras' theorem- you can figure out all of the equations of relativity. So the short answer to your question is that time dilation- which is just one of the effects predicted by the theory- is a result of the constancy of the speed of light.
Quite why light has a constant speed is still an open question. Perhaps we're in the same position as the Greeks, and one day people will find an underlying reason for it.
Time dilation is an entirely reciprocal effect. When we think time is slowing down for someone moving quickly relative to us, they think that our time is slowing down and theirs is running normally. This is a bit like perspective. If I stand far away from you I appear smaller to you, but from my perspective I'm the same size and it is you who has appeared to have shrunk.
A: Yes.
The trick behind it is to understand what does it mean to measure time and what does it mean to measure distance and once you measure it, how to communicate it to others. The key thing is to define simultaneity in such a way that makes both mathematical and physical sense, because measurement of f.e. distance means to simultaneously read data at the two points of your measuring tape or whatever you use. So you need some method to determine when the measurement was indeed simultaneous.
And so it happened, that nature gave us one nice process that allows for nice definition of simultaneity which obeys all the physical and mathematical requirements - propagation of light. 
How the light propagates, however, must be taken as fact of nature, that has no deeper mechanism. But once you have this, the supposed slowing down of time is just due to how different observers look at what time is. 
There is no mechanism however, because there is really no notion of time that is intrinsic to nature. The nature has different structure - that of minkowski spacetime, but does not have structure of space and time. The fact that we get used to the dividing spacetime on space and time is "our problem". Nature is not that way and that leads to all the "difficulties" we get when we try to do it outside of our day-to-day experiences.
A: I don't have enough reputation to add a comment but I would like to point out that even without Einstein's relativity and in an absolute frame of reference all inter-particle communication happens slower and slower as velocity increases, if you consider communication as message exchanges between particles at the speed of light. This would result in outside observers experiencing mass increase in the accelerated object and the object experiencing its own time as slower .
A: All this space time talk is unnecessary and confusing.  The concept of time slowing down means the frequency of the sound or radio/light wave you are using to communicate appears to increase when 2 objects are coming together and decrease when they are moving apart.  Suppose Nolan Ryan (former Texas Rangers pitcher) is standing a mile away from you and throwing a ball toward you once per second. (He had to eat an extra helping of cheerios to perform that feat). The frequency is once per second.  However, you don't catch the first ball until a second later, so we have a combination of frequency and distance involved here.  If he is standing still, the frequency will be once per second.  But if he is standing up strapped to the back of a pickup truck going sixty miles per hour (a mile a minute) and throws the first ball exactly one mile from you, the ball will be moving 88 feet per second faster than it was when he was standing still.  Also when he throws the second ball one second later, the ball will have to travel 88 feet less that one mile, so the additional speed of the ball and the decreased distance it has to travel means the ball will reach you sooner than one second, and the same with the next ball. This results in an increased frequency of 5280 feet (5280 feet in a mile) divided by 176 (88 feed faster, 88 feet shorter) or 1.034 balls (pulses) per second compared to 1 pulse per second if Mr. Nolan were standing still.  My calculations may not be precise, but that is the principle.  So when a car passes you, the frequency appears higher when it approaches you, then immediately appears lower as it passes.  If you were receiving radio communications between 2 objects approaching each other - whether from sound or radio waves, the sound frequency would appear to increase from both sides; it would also appear to decrease when the objects travel apart.  Virtually everyone has experienced this.  A logical conclusion to this is that if you leave earth in a space ship, the time (radio frequency) would appear to decrease, but when you returned to earth the time (radio frequency) would appear to increase correspondingly so you will not have lost even a microsecond.  So Planet of the Apes is pure scientific fiction.  Not really difficult to understand.  When a physicist says that time slows down or speeds up between two objects moving at different speeds, the physicist is simply replacing a clear explanation with a confusing scientific misnomer.  If I can confuse you, I must be smarter than you.
The e=mc2 formula is also just as simple to explain.  I like to think of Mr. Einstein as a Master of Illusion.
