Why do people use stabilisers?

The subject of stabilisation is fairly involved.  For example, if the bow is held in a vice like grip and the arm does not move when the arrow is released there is no need for stabilisers at all.  Such a technique would rapidly lead to fatigue if attempted and is never used except by robots in life tests.  In practice, the bow is held fairly loosely so there is always a tendency to allow the bow to move and twist as it leaves the hand.  The purpose of stabilisation is to reduce the amount of this movement while the arrow is on the string - what happens after that is irrelevant.

After the arrow is released it stays on the string for about 15 milliseconds.  The way the bow moves during this period is critical.  By adding a large mass to the bow it is possible to reduce the movement left, right, up or down but any torque due to the hand action must be counteracted by increasing the moment of inertia in all the axes.

The bow has a low moment of inertia about the vertical axis so that any torque variation on the handle will cause a horizontal spread in arrows.  For this reason the most important stabiliser is the long rod.  This gives an increase of inertia in the required axis as well as some added increase tending to reduce up/down spread.  Any other stabilisers are to increase the inertia so as to reduce the tendency of the bow to tilt sideways during the critical period.  This would also cause arrows to disperse left/right.

To maximise the effect of any stabilisation, the torque applied by the bow hand should be minimised.  The bow hand must be open, not sticking to the bow and the bow being free to leave the hand.  Thus a sling of some sort is essential to get the best from any stabilisation set-up.

A discussion of the optimum set-up assuming a balance is needed between bow poundage and physical weight can be found in the linked file.