Il circolo di lettura sceglie il libro del mese.

Because Italian uses the definite article il before both masculine singular nouns here:

• il circolo = the club / the circle
• il libro = the book

Unlike English, Italian often uses articles more regularly, so it is completely normal to see il repeated in the same sentence.

Literally, circolo di lettura means reading circle:

• circolo = circle, club, group
• di lettura = of reading / for reading

In natural English, this is usually understood as book club or reading group.
So il circolo di lettura is a set phrase meaning a group of people who meet to discuss books.

Italian often uses di + noun where English might use another noun as an adjective.

So:

• circolo di lettura = reading club
• literally: club of reading

This pattern is very common in Italian. A few similar examples are:

• sala da pranzo = dining room
• scarpe da tennis = tennis shoes
• libro di storia = history book

Sceglie is the third-person singular form of the verb scegliere (to choose) in the present tense.

Here the subject is il circolo di lettura, which is singular, so the verb must also be singular:

• io scelgo = I choose
• tu scegli = you choose
• lui/lei sceglie = he/she chooses
• il circolo sceglie = the club chooses

The form scelge is not correct.
With verbs like scegliere, the stem changes in some forms, so it is something learners usually just have to get used to.

Because the grammatical subject is il circolo, which is singular.

Even though a club contains many people, Italian treats circolo as one unit:

• Il circolo sceglie... = The club chooses...

This is similar to English when we say the team chooses or the group decides, though English can sometimes vary more with collective nouns.

Del is a contraction of di + il:

• di = of
• il = the
• del = of the

So:

• il libro del mese = the book of the month

This kind of contraction is very common in Italian:

• di + il = del
• di + lo = dello
• di + la = della
• di + i = dei
• di + gli = degli
• di + le = delle

Italian normally uses the article here: il libro del mese.

English can sometimes drop articles in labels or titles, but in normal Italian syntax the article is usually included. So il libro del mese is the standard way to say the book of the month.

It can mean either, depending on context.

The Italian present tense often covers both:

• a present action: The book club is choosing the book of the month
• a habitual action: The book club chooses the book of the month
• a general statement: The book club picks the monthly book

Without more context, the sentence is usually understood as a general or habitual statement.

Yes, absolutely. That is probably the most natural English translation in many contexts.

Even though circolo di lettura is literally reading circle or reading club, book club is often the best natural translation.

The normal Italian word order is:

subject + verb + object

So:

• Il circolo di lettura = subject
• sceglie = verb
• il libro del mese = object

Italian word order is somewhat flexible, but this is the most neutral and natural order.
You could change it for emphasis, but for a learner, the original order is the safest and most standard.

A learner often finds this word tricky.

Roughly, sceglie sounds like SHEL-yeh, though that is only an approximation.

A few useful points:

• sc before e or i sounds like sh
• gli in many words is a special Italian sound, something like a soft ly
• so sceglie is not pronounced like English skeg-lee

If you want to say it carefully, think something close to:

sheh-lyeh

But the real Italian gli sound takes practice.

Yes, in many contexts you can hear club del libro or club di lettura too.

However:

• circolo di lettura sounds very natural and fully Italian
• club di lettura is also understandable and common in some contexts
• club del libro is often very close to English book club

The best choice depends a little on style and region, but circolo di lettura is perfectly good standard Italian.

Because il libro del mese is the direct object of sceglie.

The verb scegliere takes a direct object directly:

• scegliere un libro = to choose a book
• scegliere il libro del mese = to choose the book of the month

So no extra word like a or per is needed.

It breaks down like this:

• Il circolo di lettura = noun phrase, the subject
• sceglie = verb
• il libro del mese = noun phrase, the direct object

So the pattern is:

Subject + Verb + Direct Object

More specifically:

• Il = the
• circolo = club/circle
• di lettura = reading
• sceglie = chooses
• il libro = the book
• del mese = of the month

This is a very typical and useful Italian sentence structure.