In the early part of the 20th century, medical science and anthropology had progressed to the point where the idea of the brain as the seat of intelligence and the theory of evolution were widely known, if not always accepted. One of the most popular conceptions of evolution was "the great chain of being," where it was understood that there was a progression in evolution - from the ape to homo erectus (or Piltdown man) to homo sapiens - with brains getting bigger and bigger to show increased intelligence. This was in part the idea behind phrenology, facial angles, and other early physiometric disciplines - and the associated scientific racism which attempted to show that certain peoples were "less evolved" due to having smaller braincases or retaining primitive traits.

In science fiction, big brains (and, increasingly, minimized physical forms) became a visual shorthand for intelligence (and later psionic powers, etc.) So the bulbous-headed alien derives more-or-less directly from old pulp fiction cover art, and you can see variations of it still in common use - such as Professor Xavier being bald, or the Talosians from the Star Trek episode "The Cage" (1964).

The particular image of "Little Green Men" was not automatic; but present from a very early period. H. G. Wells for example postulated in "Man of the Year One Million" (1893):

The coming man, then, will clearly have a larger brain and a slighter body than the present. [...] Then what is not needed siappears. What use is there for external ears, nose, and brow ridges now? [...] Directing his thoughts in this way, the reader may presently conjure up a dim, strange vision of the latter-day face: "Eyes large, lustrous, beautiful, soulful; above them, no longer separated by rugged brow ridges, is the top of the head, a glistening, hairless dome, terete and beautiful; no craggy nose rises to disturb by its unmeaning shadows the symmetry of that calm face, no vestigial ears project; the mouth is a small, prefectly round aperture, toothlesss and gumles, jawless, unanimal, no futurile emotions disturbing its roundness as it lies, like the harvest moon or the evening star, in the wife firmament of face."

Wells used a similar depiction for the Selenites in The First Man in the Moon (1901). But of course of some of the earliest aliens were truly alien. In The War of the Worlds (1897), Wells posited the Martians as tentacled cephalopods; but the unofficial sequel Edison's Conquest of Mars (1898) have large heads and bulbous eyes, and the concept of the "bug-eyed alien" would be popular in pulp fiction covers into the 1950s. "Little Green Men" as a phrase was current by the early 1950s, although it's not exactly clear when or why it gained popularity - there were already plenty of usages in the pulps, and it began to percolate off into newspapers and other non-fiction sources, including film - look at the aliens in Invasion of the Saucer Men (1957) as one of many examples.

One of the spin-offs of the science fiction pulps were publications like Fate Magazine which specialized in "real" scientific theories and encounters - this fed the burgeoning market that was interested in the lost continent of Atlantis, the Shriver Mysteries, and alien abductions. These publications reported on incidents like the Roswell incident (1947), the Kelly-Hopkinsville alien encounter (1955), and the Barney and Betty Hill encounter (1961), which popularized notions of alien visitations, flying saucers, and specific depictions of the aliens - newspaper accounts of the Hopkinsville incident of a farmhouse under attack by diminutive humanoids added the "green" color. Joe Nickel gives a good account if you're curious as to the details. The Barney and Betty Hill aliens were reportedly grey-skinned, although not hairless.

I don't want to delve into speculation here, but whether or not you believe that early science fiction depictions of aliens affected later sightings and encounters, it is indisputable that they encounters in turn fed science fiction depictions of short, bulbous green- and grey-skinned aliens (and to a lesser extant also reptilians). This was popularized by films like Encounters of the Third King (1977, greys) and the Great Gazoo in The Flintstones (little green man, first appeared 1965). The popularity spread in the 80s and 90s with far too many depictions to count, although one of the most popular and influential for "realistic" Greys was the Roswell Alien Autopsy video hoax (1995).

From there, the bulbous-headed, diminutive, big-eyed grey-or-green alien became essentially a cartoon or mascot figure. Hence 👽.