Ideally, we would prefer to find adoptive homes for stray cats, but sometimes strays are too afraid of people to be adopted, and sometimes the shelters are full and there are too many cats to find homes for them all.
In this situation, we trap the cat, have it spayed or neutered, and then release it back into the wild. At the very least, this curbs the cycle of cats having kittens. But it also directly benefits the altered cats: males fight less and females do not have to go through the stress of pregnancy.
We always use humane Havahart-brand traps. They do a good job of catching the critters without hurting them.