Þegar skjávarpinn byrjaði loksins að virka, sáum við allar glærurnar skýrt.

Þegar introduces a time clause and means when here.

So Þegar skjávarpinn byrjaði loksins að virka is a subordinate clause meaning something like when the projector finally started working.

A very common pattern in Icelandic is:

Þegar + clause, main clause

So this sentence is built as:

• Þegar skjávarpinn byrjaði loksins að virka = when the projector finally started to work
• sáum við allar glærurnar skýrt = we saw all the slides clearly

Because -inn is the suffixed definite article in Icelandic.

• skjávarpi = a projector
• skjávarpinn = the projector

Here it is the subject of the þegar-clause, so it appears in the nominative singular form:

• base noun: skjávarpi
• definite nominative singular: skjávarpinn

This is very normal in Icelandic: instead of a separate word for the, the article is usually attached to the noun.

This is a very common verb pattern:

• byrja = to begin / start
• að virka = to work / function

So byrjaði að virka means started to work or began functioning.

The word að is the infinitive marker here, like to in English:

• virka = work, function
• að virka = to work, to function

And byrjaði is the past tense of byrja, because the sentence is talking about a completed past event.

Loksins means finally.

In this sentence it modifies byrjaði að virka, so the idea is:

the projector finally started working

Its placement is natural and idiomatic. Icelandic adverbs like loksins can sometimes move around a bit, but this position is very common.

So:

• skjávarpinn byrjaði loksins að virka = the projector finally started to work

This is because of Icelandic V2 word order in main clauses.

In a normal main clause, you might say:

• Við sáum allar glærurnar skýrt.
  We saw all the slides clearly.

But when something else comes first — here, the whole þegar-clause — that first element takes the first slot, and the finite verb comes next:

• Þegar ..., sáum við ...

So the structure is:

1. first element: Þegar skjávarpinn byrjaði loksins að virka
2. finite verb: sáum
3. subject: við

That is why við comes after sáum.

Sáum is the 1st person plural past tense of sjá = to see.

So:

• sjá = to see
• sá = saw
• sáum = we saw

It is an irregular verb, so the past tense does not look like the infinitive stem.

In this sentence:

• sáum við = we saw

Because allar has to agree with glærurnar, and glærurnar is definite.

• allar = all
• glærurnar = the slides

Together, allar glærurnar means all the slides.

Here is what is happening:

• glæra is a feminine noun
• glærur is plural
• glærurnar is the slides
• allar is the form of allur that matches feminine plural accusative

So both words show grammatical information:

• allar agrees in gender, number, and case
• glærurnar carries the definite article

It is in the accusative, because it is the direct object of sáum.

The verb sjá normally takes an accusative object:

• sjá eitthvað = to see something

So here:

• sáum = we saw
• allar glærurnar = all the slides

A helpful detail: the noun form glærurnar looks the same in nominative and accusative plural, so the clearest visible clue is actually allar, which shows the accusative plural feminine form.

Because skýrt is functioning as an adverb, not as an adjective describing the slides.

The sentence means:

• we saw the slides clearly

not

• we saw the clear slides

In Icelandic, the neuter singular form of an adjective is often used adverbially:

• skýr = clear
• skýrt = clearly

So skýrt modifies the verb sáum and tells you how we saw the slides.

That is why it does not agree with glærurnar.

The comma is standard and helpful here because the sentence begins with a subordinate clause:

• Þegar skjávarpinn byrjaði loksins að virka, sáum við allar glærurnar skýrt.

The comma marks the boundary between:

• the time clause
• the main clause

In careful written Icelandic, this comma is normal in a sentence like this.

The difference is definiteness.

• allar glærurnar = all the slides
• allar glærur = all slides

So allar glærurnar refers to a specific set of slides already understood in the context, while allar glærur sounds more general.

In this sentence, the definite form makes sense because the speakers are presumably talking about the slides in that presentation.