Why? I run games in FR because it is a near-endless spring of ideas. Many brilliant designers, creators, novelists and worldbuilders have labored on FR, each adding a twist or bit, and now I'm spoiled not only for adventure ideas but by variations of the campaign setting. There's a kernel in each major character, nation, faction, or region that will interest you, even if you discard the rest.
How do you approach the setting? What I like is in; what I don't is not. I twist, change, pick and choose at my own liking. That way, the long history ceases to become a chain and becomes a toolbox. I lean into the aspects that interest me, steal from other settings, change bits here or there that I think make it more evocative, etc. I do forbid my players from browsing the FR wiki and enforce such forbiddance stringently; any knowledge they get about the setting comes from me.
How long has it been your campaign setting of choice? As a dungeon master, probably going on about 6 years. I used to homebrew, then I used other published settings, and eventually I settled on FR.
What are your favorite game supplements? If we are sticking to 5e, Waterdeep: Dragon Heist has been my favorite adventure to run, ever. What a blast; the book is so full of hooks one could play it forever. I make extensive use of 5e's adventures as toolboxes, especially the ones published during Mike Mearls' time as head of the D&D team. Each adventure doubled as a very well-realized mini campaign setting, and that is probably a testament to the team but especially to Chris Perkins, who wrote most of them.
Despite the disappointment that is the SCAG, we have had some sourcebooks that are mostly FR-centric and I do use extensively, but could also be used in other settings: Volo's Guide to Monsters and Mordenkainen's Tome of Foes have fantastic chapters at the front that really help me bring peoples, monsters and different races to life. I love the random tables for hags, yuan-ti, etc., and I adore the discussions of elven metaphysics, dwarven clans, etc. These chapters were later sadly deleted in the reprinting, but I highly recommend them.
Minsc and Boo's Journal of Villainy is also very useful and full of gameable material within, including patrons, enemies, allies, rivals, cities, etc. The DMs Guild also sells other great third-party (but not necessarily canon, if you are one to care about such things) material, including Darkhold: Secrets of the Zhentarim and Ed Greenwood's Thay: Land of Red Wizards.
As for older stuff, I really 4e's Campaign Book, which is also full of ideas despite its horrible reputation. Actually, 4e's DM-focused line is mostly pretty great if you get past the endless retcons, including FR's Neverwinter and Mezoberranzan books. 3e had great supplements too (Powers of Faerûn was underrated), and of course 2e spoiled us with so many releases I probably haven't even glimpsed them all.
Do you use any of the novels? As inspiration, yes. I've read Salvatore, but what I found most useful were Erin Evans' books (Brimstone Angels), which feature tieflings, dragonborn, and warlocks. It really solidified some of those narrative beats in my mind. I mine novels for characterization, setting, and ideas, but I do not abide fully by its continuity at my table. My entertainment as a reader is separate from my worldbuilding as a dungeon master, and sometimes heroics in the novels delete adventure hooks best saved for the player characters.