In Southwerk at the Tabard as I lay
Redy to wenden on my pilgrymage
To Caunterbury with ful devout corage,
At nyght was come into that hostelrye
Wel nyne and twenty in a compaignye
Of sondry folk, by aventure yfalle
In felaweshipe, and pilgrimes were they alle,
That toward Caunterbury wolden ryde.
From a basic utility standpoint, inns and taverns were a common gathering-place for travelers of all stripes and stations to meet, either by chance or by design, and often a focal point for smaller communities. That's why in The Lord of the Rings (1954) when the Hobbits arrive at Bree, they meet Strider at the sign of the Prancing Pony.
Inns also served as a relatively common element in other works that inspired Gary Gygax and Dave Arneson in the creation of Dungeons & Dragons (1974), perhaps most notably the the Fafhrd & Grey Mouser stories published by Fritz Leiber, Jr. often involve them drinking and carousing in inns and taverns, and he might have picked up on the idea from Robert E. Howard's "The Tower of the Elephant" (1933) which opens:
TORCHES flared murkily on the revels in the Maul, where the thieves of the east held carnival by night. In the Maul they could carouse and roar as they liked, for honest people shunned the quarters, and watchmen, well paid with stained coins, did not interfere with their sport. Along the crooked, unpaved streets with their heaps of refuse and sloppy puddles, drunken roisterers staggered, roaring. Steel glinted in the shadows where wolf preyed on wolf, and from the darkness rose the shrill laughter of women, and the sounds of scufflings and strugglings. Torchlight licked luridly from broken windows and wide-thrown doors, and out of those doors, stale smells of wine and rank sweaty bodies, clamor of drinking-jacks and fists hammered on rough tables, snatches of obscene songs, rushed like a blow in the face.
In one of these dens merriment thundered to the low smoke- stained roof, where rascals gathered in every stage of rags and tatters—furtive cut-purses, leering kidnappers, quick- fingered thieves, swaggering bravoes with their wenches, strident-voiced women clad in tawdry finery. Native rogues were the dominant element—dark-skinned, dark-eyed Zamorians, with daggers at their girdles and guile in their hearts. But there were wolves of half a dozen outland nations there as well. There was a giant Hyperborean renegade, taciturn, dangerous, with a broadsword strapped to his great gaunt frame—for men wore steel openly in the Maul. There was a Shemitish counterfeiter, with his hook nose and curled blue-black beard. There was a bold- eyed Brythunian wench, sitting on the knee of a tawny-haired Gunderman—a wandering mercenary soldier, a deserter from some defeated army. And the fat gross rogue whose bawdy jests were causing all the shouts of mirth was a professional kidnapper come up from distant Koth to teach woman-stealing to Zamorians who were born with more knowledge of the art than he could ever attain.
So these were the kind of examples that Gygax & Arneson were drawing off of, and it was clear that early in the life-cycle of the game there wasn't a strong focus on "you all meet at an inn," because the main reference we can read in the white box rules from 1974 has it as:
RUMORS, INFORMATION, AND LEGENDS: Such activity as advertising will certainly gain the notice of the locals and begin a chain of rumors. So will almost any other unusual activity. Even the departure of a party from a town is likely to be noticed. Obtaining such news is usually merely a matter of making the rounds of the local taverns and inns, buying a round of drinks (10–60 Gold Pieces), slipping the barman a few coins (1–10 Gold Pieces) and learning what is going on. Misinformation is up to the referee. Legends will be devised by the referee as the need arises, but they are generally insinuated in order to lead players into some form of activity or warn them of a coming event.
So initially in D&D, the inn/tavern was more of a place to gather adventure hooks and information than a place for the party to assemble for the first time. But it quickly became the case, particularly in professionally published modules, to start at a tavern or inn - as in Tomb of the Lizard King (1982) - and the basic idea worked well enough for a lot of roleplaying games that came after Dungeons & Dragons that it acquired it's status as a literary trope. For example, in many Shadowrun (1989) games, the players are professional criminals called shadowrunners who often meet their prospective employer ("Mr. Johnson") in a bar or restaurant, because it's a reasonable public place to meet, even if they move somewhere more private for any actual planning. You can see the same mentality at work in the opening to the film Ronin (1998).
Overuse has made it a bit of a cliche, but the general idea of "We need someplace for all these different people to meet together and hey here we have someplace where lots of different people to meet together" is so useful that it still finds its way into plenty of media, even if it isn't always explicitly in the "You all meet in a tavern and are hired for a job" sense.