I want to expand a bit on Slarty's answer.
First, I'd like to second his emphasis on just how much rainfall you'd need; it's much more than you'd think. Something that most people don't realize is just how much water there actually is in soil. One of the primary reasons that deserts get so hot in the day, and so cold at night is actually because there is no water in the soil. It's just like how being close the the ocean moderates the climate of a region, because the immense thermal mass of the ocean slows down the temperature fluctuation. Well, the large amount of water in the soil actually does the same thing; although not quite as strong as the ocean. The thermal mass of water is huge, compared to most materials. Water has a specific heat capacity of 4,182 J/Kg C, while quartz sand is only 830 J/Kg C. Check out this video of a gardener reviving dry soil; he's not quite starting with completely dead sand, but it should give you a better idea of just how much water is needed.
Secondly, as far as what humans can do to accelerate the de-desertification, again, the primary objective should be retaining water. Extremely dry soil, actually has the counterintuitive property of being hydrophobic (i.e. it repels water). So, in areas with very dry soil, you may get a monsoon, but the extreme vast majority of that water will just skim across the surface, and virtually none of it will get absorbed. What you need is either frequent soft rains, or something to get the water to stagnate (or at least slow down) over the land, to give it a chance to soak through the initial hydrophobic property of the dust, and get into the soil. And even once you have gotten some water into the soil, you need to keep it there. If the air is very dry, it will suck the water out of the soil, and all that water will evaporate and be lost again. Biochar is a very effective method of retaining water in soil, and it can even be done with extremely primitive technology (ex: clay or dirt mounds on top of a wood pile). Basically, you just burn or heat up wood, in a container with zero or minimal oxygen, and it turns into charcoal, but the microscopic structure of the wood is retained, which produces tiny little pockets with immense surface area. You grind the charcoal up into the consistency of sand or smaller, and mix it in with the soil. All these tiny particles act like little sponges, and protect the water from evaporation, thus allowing soil to retain water far longer than normal. Also, biochar has profound benefits for soil health, as it gives micro organisms a safe place to hide, and it can retain soil nutrients to prevent them from being washed out of the soil. When combined with things like rock dust (either natural, as would be present in dessert dust, or manufactured), it can significantly increase the long term health & productivity of the soil, as well as the speed at which a soil can regrow fertility. There are several ancient sites across the world, where peoples were using biochar thousands of years ago. You don't necessarily have to know the science in order to discover that it works.
After water, the biggest thing is getting organic material back into the "soil". A desert has been bone dry for so long, that basically all of the organic matter has turned to dust, and either blown away or broken down into sand and trace minerals. Most plants don't like growing in sand; they need the protection & symbiotic environment of soil, which is largely defined by the presence of organic matter. When you're doing this on a large scale, you really want to get nature to do most of the work for you, so you want to set up the conditions for the ecosystem to build the soil & organic matter for you. One way to do this, is planting trees that are hardy enough to survive the initial harsh environment. If you can get them to survive, their roots will push sugars into the soil to cultivate fungus & bacteria who in turn process the raw minerals and make them available to the root systems. Once you have enough water in the soil, these fungus & bacteria can begin breaking down raw organic matter into soil. China is currently using this strategy to stop "the yellow dragon", a region of desertification that is spreading rapidly across China.
To help this process along, you can bring in organic matter to jump start this process. There are 2 ways to do this:
- On a small scale, you can bring in compost and mix it in with the native dirt. When combined with the introduction of water (and optionally biochar) this has the benefit of instantly boosting the fertility of the soil to that of semi-high grade soil (depending upon how much compost you put in), that is immediately ready for growing growing crops or high-grade vegetation. This could be appropriate for outposts, or oasis scattered throughout the area to serve as introduction sites for worms & animal species, which will eventually be required for full restoration, to speed up their migration into the new ecosystem. However, compost takes time & manpower to produce, so a cheaper alternative would be to mix raw manure into the native soil; as seen in the gardener video I referenced earlier. This has the same benefits of boosting organic matter, and speeding up the restoration process, however, especially if you're mixing it with the raw sand of a desert, it will require time to decompose (with sufficient hydration & bacterial life), before it becomes high-grade soil.
- On large scales this amount of work becomes infeasible, unless you have the cultural incentive, financial capacity, or dictatorial authority to command the necessary manpower. An alternative would be widespread mulching. Covering the surface of the ground in wood chips or straw around 1 foot deep. This acts in two ways: First, it acts as a form of water retention, as the top layer of mulch protects the bottom layer from sunlight, and partially insulates it from heat & wind which would accelerate evaporation. Second, the mulch acts as a source of organic matter. With sufficient water & fungal spores, the mulch will rot into a good quality soil. But to maintain this soil, you either require consistent application of mulch until the ecosystem is stabilized enough for planting, or biochar & hardy vegetation to stabilize it. Also, on large scales like this, of trying to transform an entire desert, none of this stuff has to be perfect. So, just going through vast areas, and spreading a 1 inch layer of biochar, with as much manure as you have access to, scattered over it, and covering that with a 1 or 2 foot thick layer of mulch (wood chips, straw, wheat stalks, etc.), with a grid of trees planted every 10 or 20 feet, would be fine, and would drastically accelerate restoration of the area. The fungal mycelium, and worms, will take care of spreading everything out & mixing it up.
Finally, another thing to keep in mind, is that once you have done this on a large enough scale, the presence of plants (especially trees), actually starts to produce more rain.
