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Because space and time are just two aspects of the same thing. Your velocity isn't a 3-dimensional vector--it's actually a 4-dimensional unit vector, and the direction in which it points is what you consider to be the future. The non-constant nature of 3-velocities that we actually measure is just a result of the non-constant nature of projections of that 4-dimensional velocity vector onto different 3D spaces--i.e., different inertial frames of reference.

To get an intuitive idea of why this necessarily results in symmetrical time dilation, imagine two people walking along non-parallel paths at a constant rate on a 2D surface. From either person's point of view, the other person has a one-dimensional relative velocity, either towards or away from the observer, and that relative velocity depends on the angle between their paths. One-dimensional acceleration is just rotation in the 2D space. Now, what happens if you project one person's path onto the other person's 2-velocity? The projection will be shorter! And remember, the direction of your velocity is the direction of forward time from your perspective. So, from your perspective, the other person has traveled less distance along the time direction than you have, because some of their constant-velocity path was used up traveling in space instead. I.e., from your perspective, time has slowed down for them. But, projecting your path onto their velocity vector also results in a shorter path--so the effect is 100% symmetrical!

Now, this analogy fails in two ways because the real universe doesn't have any meta-time that you can use to observer where the other guy is "right now", and because spacetime rotations are hyperbolic rather than Euclidean, but those two sources of error happen to cancel out nicely and you get the correct result that moving objects appear to move through time slower.