A while ago I read about this guy wjo is trying to reverse desertification in South Africa with livestock and having success with it. Basically everyone thinks that livestock is the leading cause for desertification but he proved the opposite. What actually leads to desertification is when there's seasonal rainfall, in the off seasons the grasses that grow there tend to die but not fast enough so the already dead plants block the sunlight for new grasses. Livestock will eat the dying grasses and help foster new plant growth and over time actually reverse desertification. Not an expert on this topic, so no guarantees that all of the above is correct. https://ourworld.unu.edu/en/reversing-desertification-with-livestock Here's an interview with the guy from 11 years ago.

Now the livestock method only works for areas that aren't full blown deserts yet. I can't tell you exactly how deserts have turned back into other forms of environments historically but since the main issue with deserts is that there's not enough nutrients in the ground for anything to grow that's what you have to fix. A great source for nutrients is regular rainfall, which stimulates plant growth, so as you can imagine it wouldn't be an easy fix. I imagine that if the reversing process took place in the past then it would have been after big shifts in the climate in an area. Say if it started raining regularly in the Sahara I imagine you could start growing stuff there again after a while. Maybe start with some sturdy grasses and once the soil is compacted enough you could grow bushes, and so on. Of course you still have other issues like too much sunlight.