I do not know when it came into use, but nowadays every recurved or compound target archer following the Olympic discipline uses some parts of stabilisation as soon as he/she sees fit or can afford it. Basically what the stuff does is two things: To position the centre of gravity where the archer wants it and in various ways damp out the archers releasing mistakes.
Most archers use a long so called mono-stabiliser, which is a long rod (a meter or less) protruding from just below the grip in the direction of the target. To balance this one a bit, on both sides of the handle, shorter side stabilisers (0.3 m) emerge from the same point, but point besides the archer. Also to balance the mono-stabiliser - and thus be able to move the center of gravity a bit higher up the handle - at the top of the handle another short stabiliser is used, placed on the outside of the handle, like the mono.
By using masses at the ends of these stalks, one can tune the centre of gravity to where one likes it. Usually this is near the handle's back at the top of the grip, where the archers hand pushes against it as it envelopes the grip.
The fault-damping effect can be seen by the following way of reasoning. Imagine the archer using some of his energy to move the bow in an unintended way during release. Were we to give the handle some near infinite mass, it would hardly move, and the arrow would not be disturbed. Of course, it's hard to hold a handle with near infinite mass, even without your bow-arm stretched out horizontally. In fact handles are made as light as possible, while still remaining very strong and very unbending. What people came to use was mass set out on the edge of a strong rod, mass on distance. As long as the rod is rigid enough the mass will aid the handle's mass in resisting forced rotations about a certain axis the rod was placed for. One sees that the rods are best made as light as possible while still being stiff, the mass at the outer end being the most effective in resisting handle movements. What happens when a stabiliser is not stiff enough is that the handle will rotate without the mass at the end of the rod having to move yet. This makes the rod useless for fault damping, because the arrow will be away before the mass at the end of the rod will help to resist handle movement.
I've also heard other procedures in which stabilisers and damped mass systems helped to filter certain vibrations out, giving the archer the feeling of a very steady handle during release. I wasn't convinced. I'm not sure it does any more than giving the archer the Illusion of a steady handle, I'm not sure it keeps the arrow from being influenced. I'm also not sure weather it's right to use equipment which main purpose is to damp energy, even if it doesn't consume more than just the 'bad vibrations'. As in the end it will make the bow less swift, ruining efficiency.
But hey, in this case I'd like it if someone were to prove me wrong.