Here's my attempt at trying to explain how AT (aka the AT Protocol, ATproto, ATmosphere, whatever) works in a way that will hopefully be understandable to people who aren't very tech-savvy, inspired by some other people who have also tried their hands at this recently. I imagine there really isn't one definitively best way to do this, but I do wanna make some comparisons and analogies that I feel like have been pretty underutilized so far.
Ok, so you know about the 2000s P2P file sharing days? Applications like Napster, Kazaa and LimeWire that allowed you to get some music or movies in an often not-so-legal way? Well here's that NakeyJakey video from a while back that does a pretty good job of explaining how those worked as simply as possible:
Ok so think of AT as something somewhat similar to that but applied to modern social networking services (and also with much less copyright infringement and malware).
Basically, instead of users uploading and sharing posts via a single server and application (e.g. Facebook or Instagram), they do that via multiple different servers (e.g. bsky.social, eurosky.social or selfhosted.social) and multiple different applications (e.g. bsky.app, witchsky.app or mu.social), with what server your account is on sort of acting like the computer that contains the files you want to share and the applications acting like the clients that allow you to see other people's shared files. You can change what server or application you use at any time (see PDS MOOver for the former) so that you never feel outright stuck. I would say you should choose which server you use based on who you most trust, while you should choose which application you use simply based on your own personal needs and preferences (aka whatever looks/feels the coolest to you).
And similar to how users shared files in multiple different formats (.mp3, .mp4, .exe), AT users share and create posts that are also in multiple different formats (called 'lexicons' in this case), examples being site.standard.document for blog posts (see also standard.site) and app.bsky.feed.post for Twitter-style short text posts, with each app typically specializing in one particular format, though they might specialize in many different formats as well. Additionally, this 'lexicon' system is not just used for posts but also many other things, including interactions such as follows/likes (e.g. app.bsky.feed.like or app.bsky.graph.follow) and profile information (e.g. app.bsky.actor.profile). If you already have an account on an AT server (most likely bsky.social at the moment), you can see what files/formats are already on your account here.
Coming soon (or something which may be already out depending on when you're reading this) is a way to share 'files' only with specific people, called 'spaces'; basically, spaces are created and administered by users and sort of act as, well, virtual spaces containing either people you know closely or people who all have some sort of common interest (think a broader version of modern "group chats"), and can contain any number of users you want (including just one user, which is useful for things like bookmarks).
This may have been a bit more technical than other simplified explanations, but I hope it's technical in a way that at least some people who aren't tech geeks will still be able to get a decent grasp on.