A giugno la scuola finisce, e a settembre torno a studiare in biblioteca.

With months, Italian normally uses a to mean in:

• a giugno = in June
• a settembre = in September

This is very common with months and some time expressions. English uses in, but Italian uses a here.

Examples:

• a maggio = in May
• a dicembre = in December

Italian often uses the definite article more than English does.

So la scuola finisce literally looks like the school finishes, but in context it means something like:

• school ends
• the school term ends
• classes end

Italian frequently keeps the article where English might leave it out.

Italian often uses the present tense for future events when the time is clear from context.

Here, the time expressions make everything clear:

• A giugno ...
• a settembre ...

So la scuola finisce means school ends in June, even though it refers to a future or regular yearly event.

This is very natural in Italian. English can do something similar too:

• School ends in June.
• In September I go back to studying.

Finisce comes from finire = to finish / to end.

It is the third-person singular form:

• io finisco = I finish
• tu finisci = you finish
• lui/lei finisce = he/she/it finishes

Many -ire verbs add -isc- in some forms, especially in the singular and third-person plural. Finire is one of those verbs.

So:

• finisce = ends / finishes

Because Italian usually does not need subject pronouns when the verb ending already shows who the subject is.

• torno already means I return / I go back
• so io is optional

You could say io torno, but it is usually only added for emphasis or contrast.

Examples:

• Torno a casa. = I’m going home / I return home.
• Io torno, tu resti. = I’m returning, you’re staying.

Torno comes from tornare, which usually means:

• to return
• to come back
• to go back

In this sentence, torno a studiare means:

• I go back to studying
• I start studying again
• I return to studying

So it suggests that the speaker stops studying for a while, then resumes in September.

After tornare, Italian uses a + infinitive when you mean to return/go back to doing something.

So:

• tornare a studiare = to go back to studying
• tornare a lavorare = to go back to working
• tornare a parlare = to speak again

The a is required in this structure.

Yes, in biblioteca means in the library / at the library.

Italian uses in with many places, especially when thinking of being inside a place or carrying out an activity there.

So:

• studiare in biblioteca = to study in/at the library

Other examples:

• in classe = in class
• in ufficio = in the office
• in cucina = in the kitchen

Both are possible.

• A giugno la scuola finisce
• La scuola finisce a giugno

The version in your sentence puts the time expression first to frame the sentence: In June...

This is very common in Italian. Starting with a time phrase can make the sequence of events feel clearer:

• A giugno school ends
• a settembre I go back to studying

So the word order helps organize the timeline.

It can suggest either, depending on context.

Because the sentence uses the present tense with time expressions, it can mean:

• a future plan for this year, or
• a general/regular pattern

For example:

• A giugno la scuola finisce could mean school ends in June as a normal yearly fact.
• A settembre torno a studiare in biblioteca could mean in September I’ll go back to studying in the library this year.

Italian often leaves this slightly flexible unless more context is given.

Sometimes yes, but not always exactly.

• torno a studiare = I go back to studying / I return to studying
• studio di nuovo = I study again

The difference is that torno a studiare emphasizes resuming an activity after a break, while studio di nuovo just means I do it again.

In this sentence, torno a studiare is the more natural choice because it suggests a return to a routine after summer.