So the origins of slime-like monsters can be traced back to early 20th century horror/science fiction, such as the Shoggoth from Lovecraft's At the Mountains of Madness (1931) and Joseph Payne Brennan's short story Slime (1958). These however, were less the ubiquitous blobs we hunt for exp in JRPGs and more fearful, mysterious monsters, that can change form at will, resist damage, and in general threaten the lives of the protagonists. In this setting, slimes were generally alien lifeforms or supernatural beings in a more modern setting, rather than fantasy.

So the modern "slime" as we know it was in fact first introduced by Dungeons & Dragons (1974), where they were referred to as "oozes". The author Gary Gygax drew heavy inspiration from classic fantasy and science fiction, dragons being the most obvious example, and oozes were no exception. This can be seen by how in early renditions of DND, they were described as amoeba-like primordial creatures, and only in later editions were they described more as beings of magic created by wizards. This can be interpreted as Gygax tuning the concept to fit his fantasy setting over time, simultaneously bridging the gap between classical monsters and modern slimes.

Why however did they become so ubiquitous, though? One can posit that their popularity was spearheaded by their inclusion in RPGs, which in their early forms were heavily inspired by DND and other tabletop board games. Slimes appear in Wizardry (1981), HYDLIDE (1984), and perhaps most importantly in Dragon Quest (1986), arguably one of the most influential RPGs ever created that would define the genre for decades to come.

I'd love to go further and talk about the impact of RPGs on Japanese "isekai" literature and how they resulted in the modern trend of slimes being prominent in novels and anime, such as Slime Taoshite 300-nen, Shiranai Uchi ni Level Max ni Nattemashita, but unfortunately I believe this would violate the 20 year rule, so I'll leave it here.