What you’re registering for is an account with Rebrandly, because in order to use their service to create shortlinks they require you to have an account. (Requiring users to authenticate before they can create content is quite common. You need to log in - authenticate - to this forum before you can post, for example.)
In broad terms, an API (or “Application Programming Interface”) is a mechanism for interacting with some other code.
In the case of the Requests I lesson, you will be walked through how to interact with two APIs: the Datamuse API, and the Rebrandly API.
Let’s focus on the Datamuse API.
Datamuse have created a program which contains a word list, as well as supporting information about what words rhyme with each other, what the synonyms for a given word are, what words are often used to describe another word…
To save you from having to do all that work again yourself, Datamuse have written functionality into that program which you can use to ask it the relevant questions and get a useful answer.
The functionality which allows you to send your questions (requests) to Datamuse, and which allows Datamuse to give you answers (responses), is the API. It is what allows you to interface with Datamuse’s application.
The section of the Web Development Path opens with an article about HTTP, and the process of request and response.
At this point in the course, the focus is on teaching you how to send requests and how to deal with the responses from the remote service. To stick to your restaurant analogy, it’s teaching you how to give your order to the waiter and how to go about eating your meal when it’s served.
The forums are meant to facilitate getting help from, or providing help to, other learners- just as @yizuhi has done in this case . It’s not just the mods who may have answers to questions, you know.