Die Nachbarin sagt, wir sollen nach dem Bohren noch einmal lüften, weil es im Zimmer staubig ist.

Because Nachbarin is a feminine noun. In this sentence it is the subject, so it is in the nominative case, and the nominative singular article for feminine nouns is die.

• die Nachbarin = the female neighbor
• masculine: der Nachbar

Because the subject is die Nachbarin, which is third person singular: she. German verbs change depending on the subject.

• ich sage
• du sagst
• sie sagt
• wir sagen

So Die Nachbarin sagt = The neighbor says.

German uses commas to separate clauses more consistently than English does. Here:

• Die Nachbarin sagt = one clause
• wir sollen ... lüften = the clause containing what she says

So the comma is required.

You could also say:

Die Nachbarin sagt, dass wir nach dem Bohren noch einmal lüften sollen ...

With dass, the idea is the same, but the word order changes.

Sollen often means to be supposed to, to be told to, or should in the sense of someone else's instruction or recommendation.

Here it means something like:

• we’re supposed to air the room out again
• the neighbor says we should air it out again

This is different from müssen, which usually expresses a stronger necessity: have to / must.

So sollen fits well because this is presented as what the neighbor says we should do.

Because sollen is a modal verb. With modal verbs in German:

• the conjugated modal verb goes in the normal finite-verb position
• the main verb stays in the infinitive at the end

So:

• wir sollen lüften
• wir müssen gehen
• ich kann kommen

That is why lüften appears at the end of that clause.

Because after a modal verb like sollen, müssen, können, wollen, or dürfen, German uses the bare infinitive, not zu + infinitive.

So you say:

• wir sollen lüften
• ich muss arbeiten
• sie kann kommen

Not:

• wir sollen zu lüften

This phrase means after drilling or after the drilling.

There are two useful grammar points here:

1. nach in the time sense takes the dative case.
2. Bohren is a verb used as a noun here, so it becomes das Bohren.

So:

• nominative: das Bohren
• dative after nach: dem Bohren

That gives nach dem Bohren.

Because it is being used as a noun. In German, when a verb is turned into a noun, it is capitalized.

• verb: bohren = to drill
• noun: das Bohren = drilling / the drilling

This is called a nominalized infinitive.

Noch einmal means once again, one more time, or simply again.

So noch einmal lüften means:

• air out the room again
• ventilate once more

In everyday speech, you will also often hear nochmal instead of noch einmal.

Because weil introduces a subordinate clause, and in German subordinate clauses the finite verb normally goes to the end.

So:

• main clause: Es ist im Zimmer staubig.
• subordinate clause: ..., weil es im Zimmer staubig ist.

That final ist is a very common pattern after words like weil, dass, wenn, and obwohl.

Here es is the grammatical subject, similar to English it in it is dusty. It does not refer to a specific thing like a table or a wall. It is just the normal way to build this kind of sentence.

So:

• Es ist staubig. = It is dusty.

German often uses es in general statements about conditions, weather, and situations.

Im is a contraction of in dem.

Here in expresses location, not movement, so German uses the dative:

• in dem Zimmer → im Zimmer = in the room

If there were movement into the room, you would use the accusative:

• in das Zimmer → ins Zimmer = into the room

So:

• Es ist im Zimmer staubig = the dust is in the room / it is dusty in the room
• Wir gehen ins Zimmer = we go into the room

Because staubig is a predicate adjective here, used after sein. Predicate adjectives in German do not take adjective endings.

Compare:

• ein staubiges Zimmer = a dusty room
  
  • adjective before a noun, so it gets an ending
• Das Zimmer ist staubig = the room is dusty
  
  • adjective after ist, so no ending

That is why the sentence has staubig ist, not staubige ist.