La bibliothécaire dit que le libraire du quartier ferme plus tard le vendredi.

Because libraire is a classic false friend.

• un/une libraire = a bookseller, or someone who works in / runs a bookshop
• un/une bibliothécaire = a librarian

So in this sentence:

• la bibliothécaire = the librarian
• le libraire = the bookseller

A native English speaker often mixes these up because library looks like librairie, but une librairie actually means a bookshop, not a library.
A library is une bibliothèque.

Here, la shows that the librarian is female.

The noun bibliothécaire can refer to either a man or a woman; the article usually tells you the gender:

• le bibliothécaire = the male librarian
• la bibliothécaire = the female librarian

The noun itself does not change here, only the article does.

Que introduces a subordinate clause and usually corresponds to that in English.

• La bibliothécaire dit que...
• The librarian says that...

In English, that is often optional:

• The librarian says the bookseller...

In French, que is normally required in this kind of sentence.

Because after dire que in an affirmative statement, French normally uses the indicative, not the subjunctive.

So:

• dit que le libraire... ferme... = correct

Here ferme is the present indicative of fermer.

The sentence presents the closing time as a fact or reported fact, so the indicative is the natural choice.

Du is the contraction of de + le.

• de le quartier → du quartier

Here, le libraire du quartier means something like:

• the neighborhood bookseller
• the local bookseller
• literally, the bookseller of the neighborhood

This phrase identifies which bookseller is being talked about.

Because fermer means to close or to shut.

So:

• le libraire ferme plus tard = the bookseller closes later

French often uses the active verb this way for shops, businesses, and places:

• Le magasin ferme à 18 h. = The shop closes at 6 p.m.

You could also say est fermé in other contexts, but that means is closed as a state, not the action of closing:

• Le magasin est fermé. = The shop is closed.
• Le magasin ferme à 18 h. = The shop closes at 6.

Plus tard means later.

In this sentence, it means the bookseller closes at a later time on Fridays than usual, or later than on other days.

A useful point:

• tard = late
• plus tard = later

Even though it literally looks like more late, the natural English translation is simply later.

Because le + day of the week often means on Fridays / every Friday / on Fridays in general.

So:

• le vendredi = on Fridays, every Friday, on Friday in general

Compare:

• vendredi = Friday / this Friday / on Friday
• le vendredi = on Fridays generally

This is a very common French pattern:

• Je travaille le lundi. = I work on Mondays.
• Le dimanche, tout est fermé. = On Sundays, everything is closed.

So here ferme plus tard le vendredi means the later closing happens on Fridays.

That is just how French often expresses habitual time with days of the week: definite article + singular day name.

So:

• le vendredi = on Fridays
• le mardi = on Tuesdays

French uses the singular form to express a repeated, general habit. English usually uses the plural instead.

French word order here is very natural:

• ferme = closes
• plus tard = later
• le vendredi = on Fridays

So the structure is basically:

• subject + verb + adverb + time expression

That said, French does allow some flexibility for emphasis:

• Le vendredi, le libraire ferme plus tard.
• Le libraire ferme plus tard le vendredi.

Both are fine, but the original sentence sounds neutral and natural.

In plus tard, plus is usually pronounced plu, with the final s silent.

So approximately:

• plus tard ≈ plu تار / plu tar (roughly)

This is common when plus means more or later and comes before a consonant.

By contrast, before a vowel, you may hear a z sound in liaison:

• plus intéressant → pluz intéressant

But in plus tard, the usual pronunciation has no s sound.

A French speaker would probably break it up like this:

• La bibliothécaire = the librarian
• dit que = says that
• le libraire du quartier = the neighborhood bookseller
• ferme plus tard = closes later
• le vendredi = on Fridays

So the sentence is built very straightforwardly:

[speaker] + says that + [person talked about] + [action] + [time detail]

This is a very typical French sentence pattern and is not especially formal or unusual.