Packers are just incompressible shims meant to fill the gap so that the bump rubbers (bumpstop) are contacted at the right ride height. With multiple bump rubbers they can be used to stack the rubbers and modify the force-displacement curve. All very complicated - tough to get right but easy to get wrong in reality, but very simplefied in rF2.
The reason they are used is to prevent the car from bottoming out. They both do the same thing in different ways. Usually bump rubber is compressable and can be made progressive by it's material and shape. Packers are not- just like Natureboy says. when you hit the bump rubber, your wheel rate increases drastically, but there is some spring to it. When you hit the packers, your wheel rate instantly goes to infinity.
I heard they were pretty much only used at Spa for the compression at the bottom of Eau Rouge, I have a feeling that's not true as I constantly see them on peoples setups in-game so maybe it's like that in real life too.
The packers move the rubbers. You don't hit the packers, you hit the rubbers. More packers just means you hit the rubbers earlier. The response of the rubbers is defined in the mod, so the various bump rubber responses across vehicle types should be emulated somewhat. I think there are always rubbers on the mod side, as in you have a certain amount of suspension travel left to the rubbers at 0 ride height (that travel can be negative, so you're already on the rubbers at zero ride height) and you define the characteristics (spring rate, damping rate, rising spring and damper rates [with deflection]) of the rubbers. Presumably cars without rubbers you would give very stiff settings to reflect that.
As has been previously mentioned these are to limit suspension travel during bumpstops, material and shape dependent it is chosen progressively or linearly while the packer is used only additional for distance-control and is less-close to zero compressible. The packers can be fitted from the outside on the rod of the shock absorber/damper below the bumpstops.