In a letter1 to the Royal Society of London written in 1697, Dutch microscopist Antony van Leeuwenhoek (1632-1723) gave an account of his observations of snail eggs. The eggs had been brought to him by a friend of his for Leeuwenhoek to determine which animal they belonged to. Leeuwenhoek failed to get the eggs to hatch on several occasions before it dawned on him that they needed to be prevented from drying. After he started keeping the eggs in moist soil, he was fascinated to see "Horn Snails" coming out of them2.
Leeuwenhoek identified the snails as "wine-yard snails" and noted that the diameter of their eggs was "almost the Fifteenth Part of an Inch", about 1.7 mm. An internet search for "vineyard snail" came up with 2 species: Helix pomatia and Cernuella virgata. Other snail species have probably also been called vineyard snails. The exact identity of Leeuwenhoek's snails may forever remain a mystery; one that doesn't really need a solution.
What I thought was the most significant part of Leeuwenhoek's letter was his conclusion regarding snail reproduction:
Therefore we see (as it is said before) that the generation of the Wine-yard Snail happens by Eggs, the old Opinion is consequently to be laid aside, viz. That the Snails come forth by the spoil'd and rotten Leaves of Trees, and that the Leaves of Trees, left on the Ground, produce Snails.Leeuwenhoek then went on to provide a rational explanation for the origin of the spontaneous generation myth he had just knocked down:
But we shou'd rather think, that when we in Autumn leave the Leaves of trees lying on the Ground, the Eggs of Snails lying in the Earth these Leaves lye, are better defended from the hard Cold, than those which lye where no Leaves are.A simpler and more reasonable explanation is that snails seem to come out of leaf litter because many species live and reproduce in leaf litter.
Note that Leeuwenhoek acknowledged that it had already been stated that snail reproduction took place via eggs. But he didn't bother to give a citation. Such lax acknowledgment of prior authority seems to have been a common practice at that time.
I don't know who was the 1st to figure out how snails reproduced.
1Part of a Letter of Mr. Anthony van Leeuwenhoeck, Dated Delft, Sept. 10. 1697. Concerning the Eggs of Snails, Roots of Vegetables, Teeth, and Young Oysters
Philosophical Transactions 1695-1697 19:790-799.
2By "horns" he was probably referring to the snails' tentacles.