Here's ground that had 3 or four cylces of birds on it since we've been out. Notice that it's green, looks a little dry, but overall, looks like grass. This time of year mostly foxtails and some red top, or "greasy grass" here in the Ozarks.
Walk 40 or 50 ft to a spot that hasn't had any chickens, but is the same soil type with roughly the same aspect. Boom - dead, desiccated grass galore. You can see the forage composition is mostly the same, just looks like it's been under a blow drier...You'll also notice (or if you were standing out there you would if I pointed it out), that the grass cover is much thinner here.
So how does raising the chickens on pasture help the pasture out? Well, the manure (yes chickens poop, quite a lot actually) is very high in nitrogen and phosphorous. In fact, the poultry industry (yes, Tyson, Simmons and the like) actually made cattle farming possible in the Ozarks. Our soils were so poor that before litter from the chicken houses, it used to be around 10 acres to support a cow in a lot of places.
The manure directly adds organic matter to the field, which in turn attracts good critters like worms and dung beetles (which I've seen in poultry manure!) which take the fertility into the soil profile. It also stimulates the growth of grass, which as it grows, builds up massive root systems. When the grass is grazed by the birds or deer, trampled by the birds, or otherwise dies, a good portion of that chunk of roots dies adding more organic matter to the soil. Why's organic matter so important? Well, it acts as a long-term source of fertility, slowly breaking down over time and feeding plants, and more importantly these days, by acting as a moisture sponge that keeps the soil more moist for longer.
One thing two with the birds is that they tend to lay a pretty good "sheet" of manure on the grass, which forms a really good mulch. Over time/when it rains, this breaks down as well and creates a perfect seedbed for new little grasses and other plants to explode through the temporary chicken-induced mulch, fueled by the nitrogen and other nutrients ready available once we get a good rain. The trampling the birds do is also important because it gives new plants a niche to pop up and get established since the existing sward is broken up.