I'm quite certain a lot of the points I'm making have been said before, but I'd like to try and go beyond the "kitchen sink appeal" statement I see as the most prominent of explanations.
Reason #1: Forgotten Realms is the closest thing we have to a superhero setting
Elminster Must Die by Kekai Kotaki
The personalities of the Realms are a draw for many, as evidenced by best-selling novel series chronicling their adventures, akin to how the Marvel and DC Universes revolve around the plots and conflicts of powerful men and women in a world of mortals. In spite of its high-magic, high-powered reputation, most folk of Faerûn are still in normal medievalesque professions, without a hint of magical talent. Even the legacies of Netheril and the elves tend to be relics of the past and MacGuffins to drive adventure than common-as-dirt trinkets. Even things such as Thay's omnipresent scrying network is a cause for fear, a tool for oppression, than magic which benefits all.
Even if it doesn't have explicit or even well-thought out rules for high/epic level play, Forgotten Realms is one of the few settings which is not afraid to take the kid gloves off and show what a world filled with gods and wild, powerful magic would really be like when things get dangerous. A lot of settings, both official and third-party, attempt to put a limit or veneer of low-powered "realism" or an end to the campaign when folks approach the upper levels, but Forgotten Realms rushes headlong into it without fear.
Reason #2: Dark Elves
Cover for Advanced Race Codex: Drow by Todd Lockwood
A popular enough "monstrous" race that they ended up as a core option in 5th Edition. Sure, there are many GMs sick of players with Drizz't clones or who feel that Salvatore removed the unknown mystique from ye olde days, but the fact of the matter is that a lot of players love them, whether as a PC option or as a plot element from the GMing side. No other "evil option" gets as much support or written material as they do, and I talked about this phenomenon in an earlier blog post.
Even though drow have a presence in Greyhawk and other settings, the common mental image many gamers have of them aligns closely to how they're portrayed in the Forgotten Realms. Orcs are more or less a one-note "faceless horde," goblins and kobolds low-level cannon fodder adventurers are supposed to outgrow minus some famous exceptions. Drow are appropriate for all levels, and can be a plausible threat for all kinds of schemes.
And of course, Drizz't do'Urdlen spearheaded one of the best-selling D&D novels who's still popular to this day by gamers both old and new.
Reason #3: Still room for standard fantasy
In spite of Reason #1, there's quite a bit of material dedicated to the more iconic adventure tropes. Places like the Dalelands and the Silver Marches are great starting points for low-level adventurers, with small villages in need of saving and dungeons and ruins filled with treasure for the taking! Metropolitan cities like Waterdeep had a literal underworld to explore with the megadungeon of Undermountain or the hive of scum and villainy known as Skullport.
Faerûn balances the familiarity of standard fantasy tropes which appeal to many gamers. There's quite a few adventures for low and mid-level parties published over the past decades so that things don't feel constrained to the high end of the power spectrum. Although it may not always do so well or give valid reasons why the setting's super-NPCs aren't dealing with the problem, the world does its best to accommodate adventurers who are still holding their own against common monsters on a local level as well as parties thwarting Shar's machinations to destroy the Weave itself.
Reason #4: D&D's most well-known video games are Realms-centric
Neverwinter Nights Wallpaper
Ask someone about a Dungeons & Dragons video game, and the likeliest responses you'll get back are Balder's Gate, Neverwinter Nights, Pool of Radiance, and Planescape: Torment. They may be the best known, but they're far from the only ones. In fact, there have been many D&D video games over the years, but a lot aren't well-known or aged well. Heroes of the Lance had a non-intuitive interface, Dark Sun Shattered Lands was released too late, and attempts into non-RPG playstles such as Iron & Blood: Warriors of Ravenloft met with poor reception.
With the exception of Planescape: Torment, the ones which stick out in the public consciousness are the games which use an RPG format similar to the Dungeons & Dragons ruleset and detail the people and places of Faerûn. This may not be a fact that many in the fandom want to admit, but video game RPGs overshadowed their table-top counterparts for about twenty years now. When it comes to greater nerd culture, the average geek at GameStop picking up a cartridge with the Dungeons & Dragons logo is more likely to hear the tales of Waterdeep and the machinations of Cyric than the War of the Lance or rough-and-tumble city of Lankhmar.
Reason #5: Many sourcebooks are useful, regardless of edition
Even though 3rd Edition was once remarked as "Dungeons & Dragons for bureaucrats" on an OSR forum along with similar sentiments elsewhere, the Forgotten Realms Campaign Setting for that very edition had its praises sung by old-school and contemporary gamers alike. It was a comprehensive and detailed account of the land and its people, covering races to deities and common cultural traditions as well as mechanical benefits. Even if you do not care for the D20 System and new feats and prestige classes, the wealth of system-free material was enough to justify its purchase alone.
In Conclusion: I don't think that there's any one thing which draws so many gamers to Ed Greenwood's world. I think its iconic characters, accessibility, and popular video games and novel series bolstering the table-top sourcebooks allow for an easier point of entry and engagement to share in for D&D players.