Hei ***,

Sorry to hear about your injury. Did you deck like I did?

The most important thing to know is that there is most probably a way back to climbing and a healthy body. Now it is almost 8 months since my injury and I live like I did before. The climbing is different, obviously and in fact I almost solely do bouldering now. This of course, is a 'mental' injury.
I imagine, though, that your question is about the medical side of the injury. My L1 compression fracture was a minor one and could be treated conservatively - i.e. no surgery.
The front side of the vertebrae got impressed by about 0,5cm and still is. The thing about vertebrae is that there is no way they can "grow back in shape" or something because the pressure on them is too high.
A few days after my injury there was hardly any pain in my back and just some swelling left. The following months I used to be stiff and it started to hurt when I sat for too long. Most of this was caused by a muscular deficiency, due to my back muscles being contracted like a bitch for almost a week.
The day after the accident every muscle in my body felt sore. I used to describe it by saying that it felt like 6 guys hitting me with baseball bats.
Sitting by the way is the stuff you don't want to do, because the pressure on your vertebrae is the highest. Standing or lying is the best.
I slowly started bouldering/traversing again and also PT. I now go to the gym twice a week and actually even do deadlifts and managed to pull 1.5 times my body weight just a week ago.
But you should really be careful with your fracture for about 6 to 8 weeks, and refrain from any bigger stress (for example, deadlifts) for at least 6 months.
My biggest problem in recovery were my heels, which hurt with _every_ step I took for a month, but since you did not ask about that I hope your heels are allright.

Take care and take it slowly.
Listen to your body.
Climb when you want to.

Gabriel