How do you fix a puncture on tubeless tires?
When putting tubeless tyres on I would definitely recommend that you use a sealant such as Stan's if riding somewhere with thorns. This would be your first level of defence. The sealant would seal up a thorn (or other) hole quickly and painlessly. 

You can repair a tubeless tyre with a vulcanizing repair kit, but reseating a tubeless on the trail is very difficult. It is virtually impossible to do without 'shop air' (I haven't tried CO2) so carrying a tube is important for any major leaks. 

You can simply run a tubless tyre with a tube until you get somewhere to properly fix the tyre.

Like everything else in the mtb world there are two schools of thought on whether you leave the thorn in or remove it, I would remove it. If you leave it in it plugs the hole but then if you ever do need to put a tube in you risk puncturing that also.