Bibliotekarien sa att vi skulle ge tillbaka böckerna så snabbt som möjligt efter kursen.

Bibliotekarien is the definite form and means the librarian.

• en bibliotekarie = a librarian
• bibliotekarien = the librarian

Swedish often uses the definite form when the person or thing is already known from the situation or context. So this sentence is talking about a specific librarian, not just any librarian.

Yes — both sa and sade are past tense forms of säga (to say).

• sa = very common in everyday Swedish
• sade = also correct, sometimes slightly more formal or literary in feel

So Bibliotekarien sa att ... simply means The librarian said that ...

Here att means that and introduces a subordinate clause.

• Bibliotekarien sa = The librarian said
• att vi skulle ge tillbaka böckerna ... = that we should / were to return the books ...

In English, that is often omitted:

• The librarian said (that) we should return the books ...

In Swedish, att is very commonly kept.

Skulle is the past form of ska, but here it is not just a simple past tense. It is used because the sentence reports what was said.

A very common pattern is:

• direct speech: Vi ska ge tillbaka böckerna.
  = We are to / should return the books.

• reported speech: Bibliotekarien sa att vi skulle ge tillbaka böckerna.
  = The librarian said that we should / were to return the books.

So skulle often appears in reported speech after a past-tense verb like sa.

Because skulle is a modal verb, and modal verbs in Swedish are followed directly by the infinitive without att.

So:

• vi skulle ge = we should give / return
• not vi skulle att ge

This is similar to English:

• we should return
• not we should to return

Other Swedish modal verbs work the same way:

• kan läsa = can read
• måste gå = must go
• vill äta = want to eat
• skulle ge = should give

Yes, literally ge tillbaka means give back, and that is a very common way to say return in everyday Swedish.

So:

• ge tillbaka böckerna = return the books / give the books back

There are also more formal verbs, such as återlämna, but ge tillbaka sounds natural and common in ordinary speech.

Böckerna means the books.

The singular noun is:

• en bok = a book

Plural indefinite:

• böcker = books

Plural definite:

• böckerna = the books

The ending -na is the definite plural ending here. Swedish usually puts definiteness on the end of the noun instead of using a separate word like the.

Så snabbt som möjligt is a fixed expression meaning as quickly as possible.

It breaks down like this:

• så ... som möjligt = as ... as possible
• snabbt = quickly

So:

• så snabbt som möjligt = as quickly as possible

You can use the same pattern with other words:

• så snart som möjligt = as soon as possible
• så ofta som möjligt = as often as possible
• så mycket som möjligt = as much as possible

Yes. Så fort som möjligt is also very common and would sound natural here.

There is a slight nuance:

• så snabbt som möjligt focuses more on speed
• så fort som möjligt often feels a bit more like as soon as possible

But in many everyday contexts, they are very close in meaning.

Because after att, Swedish uses subordinate clause word order.

In a main clause, the finite verb often comes in second position:

• Vi skulle ge tillbaka böckerna.

But in a subordinate clause introduced by att, the subject usually comes before the finite verb:

• att vi skulle ge tillbaka böckerna

So this sentence follows normal Swedish subordinate-clause order.

Efter kursen means after the course.

• efter = after
• kursen = the course

In the sentence, it tells you when the books should be returned:

• give back the books as quickly as possible after the course

So the idea is: once the course is over, return the books as soon as possible.

Yes, Swedish is fairly flexible here. For example:

• Bibliotekarien sa att vi skulle ge tillbaka böckerna så snabbt som möjligt efter kursen.
• Bibliotekarien sa att vi efter kursen skulle ge tillbaka böckerna så snabbt som möjligt.

Both are possible, though they may place slightly different emphasis on the timing. The original version sounds very natural.

It depends on context. All of these can work.

In this sentence, skulle can suggest:

• should return the books
• were supposed to return the books
• were to return the books

Because the meaning has already been given to the learner, the important thing is that skulle here expresses an instruction or expectation reported from what the librarian said.

So a natural English translation might vary depending on tone, but the Swedish grammar is the same.