The current edition of Dungeons & Dragons uses the Forgotten Realms campaign setting as the default location for many games, and many DMs choose to set their D&D homebrew games on the continent of Faerun in the setting. The earlier editions of D&D had dedicated Forgotten Realms books, but these were treated as supplemental material, and the setting was by no means considered the default choice. Nowadays, most games are set in Faerun, even with the numerous campaign settings on offer, and there are a few reasons for the popularity of the Forgotten Realms, compared to other D&D settings.
The early iterations of Advanced Dungeons & Dragons didn't have a set default world and the books treated most material as if it would be slotted into a homebrew game. The exception to this was basic Dungeons & Dragons, which used the Mystara setting as its default. The second edition of AD&D also lacked a default setting, but this was the era when lots of new D&D campaign settings were introduced, such as Birthright, Dark Sun, Planescape, and Spelljammer, some of which diverged from the standard fantasy setting format. In D&D 3e, the default setting was the world of Oerth from the Greyhawk campaign setting. This means that most of the campaigns and sourcebooks treated Oerth as their main setting, with notes on how to adapt certain elements to other settings. In D&D 4e, the default setting is known as Nentir Vale, which is a world that purposely lacks specific details, so that DMs are free to flesh it out as they please.
D&D 5e treats the continent of Faerun from the Forgotten Realms as its default campaign setting, though there has been a shifting focus to the multiverse at large. This might have something to do with Wizards of the Coast bringing back many of the old campaign settings, with Spelljammer set to return in August and Dragonlance returning later in the year, along with integrating planes from Magic: The Gathering into the multiverse. There are several reasons why Faerun is used as the default D&D setting, as it's familiar to non-D&D players, easy to explain to newcomers, and has lore that is easy to use or ignore as the DM needs.
The Forgotten Realms Has The Most Popular Spin-Off Media
There have been many D&D novels released over the years and they have been set across the different campaign settings, but the most popular by far is The Legend of Drizzt series by R.A. Salvatore. These books weren't the first novels set in the Forgotten Realms (that honor belongs to Douglas Niles' Darkwalker on Moonshae), but Salvatore helped flesh out the setting. The Legend of Drizzt stars Drizzt Do'Urden, a drow ranger who lives on the surface of Faerun, despite his race being shunned and despised. The story of Drizzt going from an outsider to one of the most celebrated heroes in Faerun helped the D&D novels find an audience outside of the tabletop game, and Drizzt still heavily features in D&D media, such as the Adventures in the Forgotten Realms set for Magic: The Gathering and Dungeons & Dragons: Dark Alliance. The Legend of Drizzt series started in the '80s and is still going strong today, with the next book (Glacier's Edge) launching in August. Drizzt is likely the most popular D&D character of all time and his prominence helped make the Forgotten Realms a popular choice for D&D campaigns.
The Forgotten Realms campaign setting has the most media that non-D&D players will have consumed. The majority of the D&D games are set in the Forgotten Realms, including the Baldur's Gate series, Neverwinter, Neverwinter Nights, Icewind Dale, Eye of the Beholder, and Idle Champions of the Forgotten Realms. There are a lot of people who were exposed to D&D during the 2e/3e era through the Baldur's Gate and Neverwinter Nights series, which is why the characters and locations from those games still play prominent roles in D&D media. It bears mentioning that the upcoming Dungeons & Dragons: Honor Among Thieves movie is also confirmed to be set in the Forgotten Realms, with the directors confirming during the D&D Direct that the story is set in the Sword Coast region. The fact that the Forgotten Realms was chosen over the other settings just goes to show how popular it is. It remains to be seen how much the choice of setting will matter in the movie, but there are bound to be a ton of references to the novels and video games, as well as the tabletop game.
Why The Forgotten Realms Is The Default (Not Dragonlance Or Greyhawk)
In the D&D 2e era, there were three worlds that were considered the "main" D&D settings: Abeir-Toril from the Forgotten Realms, Oerth from Greyhawk, and Krynn from Dragonlance. Oerth did become the default setting in D&D 3e, but it's easy to see why the continent of Faerun won out in the end. The classic D&D Greyhawk campaign setting is great for DMs who want a world that has the basics laid out but has enough wiggle room for them to add or remove things as needed. The problem with Greyhawk is that it doesn't have the outside appeal that the Forgotten Realms has. There have been some Greyhawk novels and video games, but they are nowhere near as popular as the ones set in the Forgotten Realms.
The world of Krynn from Dragonlance has the opposite problem. Dragonlance has many novels, with its mainline series (written by Margaret Weis and Tracy Hickman) rivaling The Legend of Drizzt series in terms of popularity and long-term appeal. The problem with Krynn is that the lore of the setting created by the novels is so dense and restrictive that it can make running a basic D&D game hard, at least not without ignoring the trappings of the setting. The world of Krynn is often involved in a number of large-scale conflicts, while some of its eras involved completely different rules for magic. Even the process of becoming a D&D wizard in Dragonlance is difficult, as the lore of the setting mentions that all spellcasters have to make this epic journey and perform a deeply personal (and dangerous) test, in order to be recognized by the Wizards of High Sorcery and not hunted as a renegade. That kind of element can introduce some fantastic character-building moments in a novel, but it's likely to derail a campaign unless everyone is willing to ignore that key part of the setting.
The Forgotten Realms has the perfect mix of widespread popularity with both D&D fans and those who are only familiar with the game through outside media and has a setting that is easy to use for all types of games, without the lore becoming too restrictive. Faerun is a place that is big and familiar, where characters like Elminster or Volo could appear to give the players a brief word of encouragement, before seeing them off on their next big adventure. This is the appeal of the Forgotten Realms and why it was chosen to be the face of the current edition of Dungeons & Dragons.