There is no real science behind it, its just nature. Killifish are a bit different then a lot of our aquarium fish. There are annual species then there are non-annual species, these terms are the same as with plants. Annual species hatch, grow, breed, and die in less then a year and that is there natural life cycle. A lot of the prettier killis are annuals, and their life cycle is why you don't see them for sale very much in LFS. Most you do see for sale are the non-annuals. Annuals have such a short life usually because they live in temporary waters that only exist seasonally, such as ponds in Africa. The fish breed when the water starts to dry up, eggs are usually buried. Eggs can withstand the dry season while the parents die. A lot of these species actually NEED the eggs to undergo this or they will not hatch. Once the wet season comes around the eggs hatch and you have fish in the puddle again(like magic!). This is only one method killis use, they are quite variable in their reproduction methods. A lot of trading/selling/buying amongst killi keepers is done via eggs not fish.
Basically this kit works the same as any sea monkey or triops kit. Same as daphnia cultures too. There is a packet with some eggs and dried medium, you add it to water and they hatch(like magic). You can get a decent number of fish if you upgrade the tank after about a week and keep up on WCs. None the less these fish only live a year at the very most, normally shorter. The exact species I found that was in these kits was Nothobranchius korthausae, but I wouldn't be surprised if there was some variability.
All in all the kits are very expensive, compared to just buying some eggs online.