They use a small tank of liquid nitrogen with a trigger and applicator. The vet marks the areas to be treated with a sharpie, and then puts the applicator to the leg. If I remember correctly it's 25 seconds of liquid nitrogen directly onto the skin, and often times there are up to 30 separate points to be 'treated'. The horse is heavily sedated for the procedure.
I assisted several times in the short amount of time I worked for one of the local vets. It's not a practice I agree with, but I'm thankful for the experience of seeing it first hand. I even had to take over (under the vet's supervision) when her hands would get too cold (I worked there through the winter, and you couldn't wear gloves to do it).