You do not need to read this page to get started with using git-annex. The walkthrough provides step-by-step instructions.
Still reading? Ok. Git's man page calls it "a stupid content tracker". With git-annex, git is instead "a stupid filename and metadata" tracker. The contents of large files are not stored in git, only the names of the files and some other metadata remain there.
The contents of the files are kept by git-annex in a distributed key/value
store consisting of every clone of a given git repository. That's a fancy
way to say that git-annex stores the actual file content somewhere under
.git/annex/. (See internals for details.)
That was the values; what about the keys? Well, a key is calculated for a
given file when it's first added into git-annex. Normally this uses a hash
of its contents, but various backends can produce different sorts of
keys. The file that gets checked into git is just a symlink to the key
.git/annex/. If the content of a file is modified, that produces
a different key (and the symlink is changed).
A file's content can be transferred from one repository to another by git-annex. Which repositories contain a given value is tracked by git-annex (see location tracking). It stores this tracking information in a separate branch, named "git-annex". All you ever do with the "git-annex" branch is push/pull it around between repositories, to sync up git-annex's view of the world.
That's really all there is to it. Oh, there are special remotes that let values be stored other places than git repositories (anything from Amazon S3 to a USB key), and there's a pile of commands listed in the man page to handle moving the values around and managing them. But if you grok the description above, you can see through all that. It's really just symlinks, keys, values, and a git-annex branch to store additional metadata.