This is because time is like space, and moving objects are tilted in time. It is no more mysterious than the statement that two rulers, one vertical and one tilted, have different heights. Not only that, but if you tilt your floor so that it is perpendicular to the tilted ruler, now the ruler that was previously vertical and taller is now tilted and shorter! There is no more mystery here than geometry, except that the geometry is slightly different from Euclid's in that the pythagorean theorem has a minus sign.
When you are standing still, your path in space-time is parallel to the time axis. When you are moving, your path in space-time is tilted relative to the time axis. If you go somewhere and come back, you make a triangle in space time, and the time you measure is the sum of the lengths of the two legs of the triangle that represent your path, and the time measured by someone who doesn't move is the third leg.
In ordinary geometry, the sum of two legs of a triangle are always more than the third, in relativity, it is always less (under the condition that an observer can travel on each leg). The reason for this difference is explained in this answer: Einstein's postulates <==> Minkowski space. (In layman's terms)
Because of the minus sign in the pythagorean theorem, there are special right triangles whose hypotenuse is of zero length. These triangles have a vertical side which is always proportional to the horizontal side, in other words, their hypotenuse represent an object travelling with constant speed. This speed is the speed of light.
It is just an accident of our universe that light travels at this speed. If light had a small mass, it wouldn't quite travel at c, but everything else would work out fine. But we would still have relativity exactly the same as before, and the parameter "c" that we call the speed of light would still be exactly the same. It would just get a new name, perhaps "the speed of gravity".