Advice and Trust: In the thread in the Sufficient Velocity forums Strypgia provides with supplementary notes and commentary to understand the story and the characters.
Power Girl fanfic A Force of Four is set in the Pre-Crisis DC universe. In order to help readers who may not be familiarized with those versions of the characters, Dark Mark provides a glossary listing characters, first appearances... It isn't necessary to read it to understand the history, but it's helpful.
Companion fic Code Geass Colored Memories, it's main focus is on the POV of Kaguya, Suzaku and Kallen. It also adds background stories to many characters like Colonel Kusakabe. It fleshes out other events that happen before or during the timeline of Colorless Memories, along with worldbuilding the Code Geass universe in more detail.
As mentioned above, the pages on Code Geass fanon wiki, there is information that is not stated in the main story that gives more information on the fic's universe, such as details to many of the Knightmare Frames, the thought elevators, full character list and information on each of the factions.
Evangelion 303: Grummancat often gives extra information in his DeviantArt page about the story, characters and backstory, clarifies doubts and explains how he came up with some concepts (such like Asuka's Cool Bike).
A large amount of Company0051's background is detailed in supplementary artwork, including the history and locations of the Chief's scars and the names and designs of everyone in the encampment. It's well worth checking out.
Renegade features a number of Codex entries similar to the Codex in Mass Effect that details additional notes about the setting.
This is actually a recurring trend in Mass Effect fanfiction. Particularly with Halo crossovers.
In the Nanoha fanfic Blood and Spirit, a chapter after the main story discusses how Arisa is able to maintain her existence as a ghost and how her powers work.
Most of the world building in Miracle Child takes place on the work's tumblr rather than in the story itself. This is understandable, since bogging down the work with tidbits on how the Alternian Empire works and what the various canon characters are doing would quickly leave no room for the plot.
The bonus chapter for Season Two of Calvin and Hobbes: The Series reveals Andy is homeschooled, and both him and Klein are named for their (proposed) voice actors (Andrew Lawrence and Robert Klein, respectively).
Between My Brother And Me: In-universe: May and Max learn all they need to know about dueling from a magazine called 'Duelist Monthly'...that their mom had for some reason. From it, they learned about Duelist Kingdom, Battle City and the 'who's who' in the world of dueling.
The Total Drama story, Courtney and the Violin of Despair includes a substantial Notes section. The narrative and notes also include hyperlinks to supplemental information on ancillary topics and embedded music videos for some of the compositions mentioned.
The Legend of Total Drama Island includes an extensive Notes (aka trivia) section, which mainly explains the more obscure allusions. In the Wikia-based version, the Notes also include hyperlinks to supplemental information on ancillary topics, mostly things alluded to in the story, and the story itself includes embedded music videos for most of the compositions mentioned.note The Fanfiction.Net version does not have these links, because that site does not support external links in story pages, so video links appear on the author’s profile page instead.
The tumblr on which the Ragnarok Online fanfiction Warriors of the World is posted is occasionally updated with a post tagged as "lore" giving background information regarding the many changes to the world of Midgard in order to have it work as a real world. On top of that, the author has stated that this is exactly what the tumblr is for since they got tired of having to reply to their own reviews on Fanfiction.Net.
Hivefled has a tumblr which provides additional information and worldbuilding on a lot of the characters.
After The God Empress of Ponykind was completed, the author began a series of blog posts dedicated to expanding upon Equestria and the history of the different factions present in the universe.
The author for Kitsune no Ken: Fist of the Fox has a Deviantart account where he posts drafts of story-notes, which he had set up as a way of giving some closure in case he ever decided not to update the story further.
There is a lot of supplemental material for Game Of Touhou in the author's deviantart page, and there are even unreleased voice-ins of every faction.
The official wiki for the supercrossover military fanfiction series, The Terminators: Army of Legend, contains extra content not found in the story itself, everything from operation names to casualty lists, as well as background Intel on specific locations and battles.
CertainDestiny, the author of Soul Eater: Troubled Souls, posts on his DeviantArt account (Grade-AMasterpiece) character profiles for the Original Characters that appear in the story. He also puts up glossaries that explain original concepts.
ParadoxProphet, the author of Despair's Last Resort has been making posts going into further details on her Tumblr, and intends to make other posts going into aspects of the story that couldn't be explained in the story proper. She also is writing Resort Mode chapters, which go into the unused free time events of the characters and gives a better idea of what they're like.
Cycles Upon Cycles: As is tradition with Mass Effect fics, most chapters ends with one or two Codex entries relevant to the events of the chapter. Recently the Codex entries have been replaced by brief dossiers of Shepard's crew.
The author of Young Justice: Darkness Falls and Young Justice Titans uses this pretty heavily, though that also follows somewhat in the tradition of the show introducing characters into storylines without a "proper" origin. Probably most blatantly done at the beginning of Titans, where the author gave full drawings/bios for the titans, including giving origins proper for Arrowette and Kyle Rayner without them being seen onscreen.
Oversaturated World: Some background details of the verse aren't covered in any of the fics proper, but on the group forum FAQ. There are even details from an otherwise unrelated series of oneshots and an old blog post by the author thrown in.
Pokémon: Nova and Antica: The writer has a whole section in his DeviantArt gallery dedicated to the additional elements and the Original Characters introduced in the story, meant for easy access to brush up on or familiarize yourself with the content. This includes commissioned artwork of his OCs. To a lesser extent, Pokemon Story Chronicles also serves as one.
The Dragon Age: Inquisition series Twice Upon An Age plays with the trope. The side volume The Skyhold Runner's Guide to Survival is explicitly presented as a manual, and it functions as one in-universe, but the reader doesn't really gain any additional information from it; it's more of a repository for Varric's humor than anything else. By comparison, the other side volume Across the Waking Sea does give the reader insight into details they otherwise wouldn't know.
Forum of Thrones has its own wiki, which is stated to include additional background information that has no room in the actual story, but can still be treated as canon to it.
The trope page is often frequented by the author, who uses it to establish minor details that may or may not be brought up in the story proper later on, such as the names of minor lords or the exact nature of some events.