Im August brauchen wir kein Hotel zu reservieren, weil unser Gästezimmer frei ist.

In German, months are very often used with in + dative article when talking about time. So:

• in dem August becomes im August
• im is just the contraction of in dem

So im August is the normal German way to say in August.

Also, putting Im August at the beginning is just a way to set the time frame first.

German main clauses follow the verb-second rule: the conjugated verb must be in the second position.

Here, Im August is placed first. That means the verb must come next:

• Im August | brauchen | wir ...

If you started with the subject instead, you would get:

• Wir brauchen im August kein Hotel zu reservieren.

Both are correct. The version in your sentence emphasizes the time.

Here brauchen means to need.

German often uses:

• brauchen + zu + infinitive

So:

• brauchen ... zu reservieren = need to reserve

In this sentence, the infinitive phrase is:

• kein Hotel zu reservieren

Literally, that is something like to reserve no hotel, but in natural English it means not to need to reserve a hotel.

A useful thing to notice: the object kein Hotel comes before the infinitive zu reservieren. That is normal in German infinitive phrases.

German usually uses kein to negate a noun phrase that would otherwise have ein or no article.

So:

• ein Hotel = a hotel
• kein Hotel = no hotel / not any hotel

Using nicht ein Hotel is usually not the normal choice here.

So kein Hotel is the natural way to say that there is no need for any hotel reservation.

Because Hotel is a neuter noun: das Hotel.

In the accusative singular, neuter kein stays kein:

• nominative: kein Hotel
• accusative: kein Hotel

If the noun were masculine, you would see keinen:

• keinen Tisch

So kein Hotel is correct because Hotel is neuter.

Because weil introduces a subordinate clause, and in German subordinate clauses are normally separated from the main clause by a comma.

So:

• ..., weil unser Gästezimmer frei ist.

This is a very regular German punctuation rule. English is often less strict here, but German requires the comma.

Because weil creates a subordinate clause, and in German subordinate clauses the conjugated verb goes to the end.

Compare:

• Main clause: Unser Gästezimmer ist frei.
• Subordinate clause: weil unser Gästezimmer frei ist

So the verb ist moves to the end because of weil.

Because Gästezimmer is:

• singular
• neuter
• nominative here

The possessive word unser changes its ending depending on gender, number, and case.

For nominative singular neuter, the form is:

• unser Gästezimmer

Compare:

• unsere Wohnung (feminine)
• unser Haus (neuter)
• unsere Gästezimmer (plural)

So unser Gästezimmer is correct because it means our guest room.

After sein, adjectives are usually predicate adjectives, and predicate adjectives do not take endings.

So:

• Das Gästezimmer ist frei.
• not freies

Here frei means available, unoccupied, or open for use.

It does not mean free of charge in this sentence. The idea is that the guest room is available, so there is no need to reserve a hotel.

Because German commonly forms compound nouns by writing them as a single word.

So:

• Gäste = guests
• Zimmer = room
• Gästezimmer = guest room

This is extremely common in German.

Also, the last part of a compound determines the grammatical gender. Since Zimmer is neuter (das Zimmer), das Gästezimmer is also neuter.