There is no 'technical' answer to your main question. The only answer is "It's what the project's author decided to put in the README", and unless you go and ask them directly, you will only get answers that try to guess at some unknown person's thought process
Why is this particular library special in that it requires some "git" stuff?
The project's README seems to be written with the standalone bin/obfuscate tool in mind – it's very likely that the author did not plan on it being used as a library at all!
However, it doesn't actually require some "git" stuff; that's just what the project author describes in the README. But you can always download a .zip archive of the files instead. And the project is available through Composer, too, and has had a composer.json file since the beginning.
Also, note that the author doesn't mention what the library is actually called in Composer, which I find extremely strange. (They sometimes have very non-obvious names there.)
It could be that the author assumed you will simply try the same name as in the GitHub address, i.e. naneau/php-obfuscator, or that you will find the name by reading the project's "composer.json" file. I think it is perfectly reasonable to assume that it will have an obvious name by default.
That being said, not all projects are automagically available through Composer – to make that happen, the author needs to register for a Packagist.org account and publish each project there first. Not being published is not actually 'special', it is the default state.
So in the general case, there is nothing very unusual about such a situation. Maybe the (hypothetical) developer just doesn't care about Composer, or is even not aware that it exists. Maybe the developer considers the project mainly as a command-line tool rather than a library dependency. Maybe the developer prioritizes Git because the project is under heavy development and is not in a state to have "releases". Maybe the developer, like you, doesn't have a Packagist account nor can register one.
What does "cloning a repository" mean
Roughly, it is similar to downloading the project's files as a .zip or .tar.gz archive.
As Git is a version-control tool (i.e. meant to track changes to code over time), the difference of using git clone to download its repository is that you'll immediately have all versions at hand. By default it will also give you the very latest "development" version, which has had a few changes since the 0.0.5 release happened.
I have never used "git", "cloned a repository",
It is perhaps time to start doing so. In general, although using Git is not mandatory (Hg aka Mercurial and SVN are two most common alternatives), using some kind of "version control system" is probably Item #1 of best practices when programming in any language, for any platform.
You'll find that most libraries and other projects are hosted as a Git repository, and Composer often uses Git "behind the scenes".
and I don't have a GitHub account, nor can I register one
There is a distinction between using Git and using GitHub. The former is a tool, the latter is a hosting provider. One can use Git without having a GitHub account, much like one can program in PHP without having a web-hosting account.
In any case, accounts are not required in order to download something from GitHub – certainly not for a publicly accessible repository.
