Stabilisation fills the natural cavities and pores in wood or bone with a resin, making it much stronger and almost completely resistant to deformations due to temperature and humidity changes, sort if turning it into a plastic. Apparently it also slows down the colour changes due to exposure to UV, but I'm not sure about this.
It's quite difficult to do on your own, as to my knowledge the process requires conditions (high pressure and temperature?) difficult to replicate without special equipment. You'll need to send it out to be stabilised; K&G is a recognised provider of this service.
There are products such as pentacryl that will strengthen wood, but it won't achieve this to the same effect as professional stabilisation.