Mein Lehrer meint, der Beamer habe gestern nicht funktioniert, deshalb sei die Leinwand dunkel geblieben.

They are Konjunktiv I forms, which are very commonly used in German for indirect speech or reported statements.

So the sentence is not simply stating the facts directly. It is reporting what the teacher says.

• Direct statement: Der Beamer hat gestern nicht funktioniert, deshalb ist die Leinwand dunkel geblieben.
• Reported version: Mein Lehrer meint, der Beamer habe gestern nicht funktioniert, deshalb sei die Leinwand dunkel geblieben.

Using Konjunktiv I creates a little distance: this is what someone claims/says, not necessarily what the speaker personally confirms.

Yes. Very much so.

The main reporting clause is:

• Mein Lehrer meint

What follows is the content of what the teacher says:

• der Beamer habe gestern nicht funktioniert
• deshalb sei die Leinwand dunkel geblieben

Both of those clauses are in Konjunktiv I, which is a strong clue that this is indirect speech.

Because German often allows dass to be omitted after verbs like sagen, meinen, glauben, and similar reporting verbs.

So these are both possible:

• Mein Lehrer meint, der Beamer habe gestern nicht funktioniert.
• Mein Lehrer meint, dass der Beamer gestern nicht funktioniert habe.

There is a word-order difference:

• without dass: normal main-clause order inside the reported clause, so the verb is in second position
  
  • der Beamer habe ...
• with dass: the finite verb goes to the end
  
  • dass der Beamer ... funktioniert habe

The version without dass often sounds a bit more formal or written, especially with Konjunktiv I.

Because meint refers to the teacher’s statement now, while gestern refers to the event being reported.

So the timeline is:

• now: the teacher says/thinks this
• yesterday: the projector supposedly did not work

This is normal in both German and English. Compare:

• My teacher says the projector didn’t work yesterday.

The reporting verb can be present even if the reported event is in the past.

Because the reported events are completed in the past, and in indirect speech German often uses:

• Konjunktiv I of the auxiliary
• plus the past participle

So:

• hat funktioniert becomes habe funktioniert
• ist geblieben becomes sei geblieben

This is basically the indirect-speech version of the normal perfect tense.

Because the verb bleiben forms its perfect with sein, not haben.

So in normal indicative German:

• Die Leinwand ist dunkel geblieben.

In indirect speech with Konjunktiv I:

• Die Leinwand sei dunkel geblieben.

This works just like other verbs of movement or change of state that often take sein in the perfect.

Because deshalb is not a subordinating conjunction like weil. It is a conjunctive adverb.

That means it takes up the first position in the clause, and the finite verb must still come in second position.

So the pattern is:

• deshalb = position 1
• sei = position 2
• die Leinwand = then the subject

So:

• Deshalb sei die Leinwand dunkel geblieben.

Compare:

• Deshalb ist die Leinwand dunkel geblieben.
• ..., weil die Leinwand dunkel geblieben ist.

With weil, the verb goes to the end. With deshalb, it does not.

It means remained dark or stayed dark.

The verb bleiben plus an adjective often means that a state continued:

• ruhig bleiben = stay calm
• offen bleiben = remain open
• dunkel bleiben = remain dark

So die Leinwand dunkel geblieben suggests that the screen did not light up or show an image. It emphasizes that it stayed in that dark state.

That is slightly different from simply saying war dunkel, which would only describe the state, not the idea of continuing in that state.

Yes. In everyday German, der Beamer means a video projector or digital projector.

This is a classic false friend for English speakers, because English does not normally use beamer with that meaning.

So in German:

• Beamer = projector

Yes. In everyday spoken German, many people would simply use the normal indicative forms instead of Konjunktiv I:

• Mein Lehrer meint, der Beamer hat gestern nicht funktioniert, deshalb ist die Leinwand dunkel geblieben.

A version with dass is also possible:

• Mein Lehrer meint, dass der Beamer gestern nicht funktioniert hat und dass deshalb die Leinwand dunkel geblieben ist.

The original sentence with habe and sei sounds more formal and more typical of written reported speech, journalism, or careful narration.