Stabilizing is the process of impregnating handle material such as wood, bone, or ivory with a hardenable resin to make it harder and not as subject the the effects of temperature and/or water. There are also some woods that cannot be used for handles without the process because they are too soft on their own, such as spalted woods. Be aware that not all woods and other material need to be stablized to be useful as handle material and there are even some woods that are so dense and oily that they cannot be. It is really not the must do thing that some people make it out to be but it can be a useful tool. Things like wood, bone, antler, horn, and ivory have been used for knife and sword handles for millennia without being stabilized and have held up just fine. For most makers it is better to send material out to be stabilized unless you are going to do large batches on a regular basis. However, if you want to do it yourself there are sites that sell the material and equipment that you will need. I believe a product called Cactus Juice is a product that has gotten good reviews.
Doug