Usually when my platys are about to give birth, the females belly has been blown up like a balloon for over a week. They tend to hide in the back behind the plants. As they start to give birth they swim upward toward the surface then dart downward toward the gravel over and over again, often pausing as each baby is released. If it is a huge batch of babies the mother will often take a break between groups, and the entire process may take several hours for them all to come out. If she is frightened during this time she will often stop for a while until she feels safe again. I've seen the babies come out anywhere from 1 to 4 at a time. The smallest batch of fry that I have seen is 6 and the largest must have been about 80 (best guestimate hard to count that many). If the mother is sick, overly stressed or not quite fully mature just before she gives birth there will be a higher percentage of still born fry.
I suggest that if you are trying to keep as many fry as possible that you move the mother to a seperate tank by herself just shortly after you notice her belly starting to bulge, and be patient and wait. I strongly discourage putting the mother of any of those breeders nets/cages sold at the pet store, because they are highly frightened and stressed by the tiny cage and fight non-stop to escape. I'd use no smaller than one of those cheap 2.5 gal tanks to put the mother in with plenty of places for her to hide, located in an area of the house where she won't have a ton of external things moving around the tank stressing her.
If you do not have the means to support a large population of platys, you may simply let nature take its coarse in your main tank, where the weak, stupid or unlucky fry are eaten by the other fish and only a few of them survive. Another thing you can do to control the population is to keep the males and females seperate. Keep in mind that once a female platy has been mated she can get pregnant over and over again every month for several months even without mating again.
If you lack extra tanks, the fry can be kept in those breeder net/cages until they are large enough not to be eaten (just larger than the mouth opening of the larger fish).
Breeding platys can be a lot of fun, just be careful not to over do it, because their numbers can quickly overwhelm you, and giving away or selling the excess is sometimes easier said than done. Also keep in mind that many pet stores and other aquarium keepers are picky and don't necessarily want your ugly mutt rejects, they want healthy, pretty, colorful, and active fish. (You know the ones you'd rather keep for yourself).