It's misleading to think of the mass as increasing (and many modern textbooks in relativity do not use the concept of "relativistic mass" because of the confusion it causes).
So the modern-day answer to "why does mass increase" is: "It doesn't". It's more useful to think of an object's ENERGY as increasing. You increase its energy in relativity just as in the old-fashioned way, by doing work on the object.
The amount of energy in an object is related to concepts of space and time. This is true even in pre-relativity physics; because a swiftly-moving rock (one that covers a large space in a small time) has more energy than an identical rock that is moving slowly.
But the formula that relates speed to energy is different in relativity, because of its different picture of how space and time are related. In pre-relativity physics, an object gains speed proportionally to the square root of the work you do on it. That means that (we thought) you could increase an object's speed to any given value, just by pouring a sufficient amount of energy (work) into it.
But relativity says that energy, space and time are related in a different mathematical way; so that in fact as you do more and more work on the object, its speed asymptotically approaches a fixed limit (which is of course "c").
This different mathematical relationship means that an object "behaves as if" it had more mass as its speed increases. It might be better to say it has more "inertia," or resistance to change; in particular, as its speed approaches "c", it becomes increasingly difficult (requires ever more work) to accelerate the object. The object seems to be more "sluggish," as though its mass has increased. And in some equations, it's useful to plug in the so-called "relativistic mass" and treat it as though it were "real" mass. But this is more in the way of a mathematical trick; nothing about the internal nature of the object changes.