Ljósið slokknar þegar rafmagnsleysi verður.

Because -ið is the suffixed definite article here.
So:

• ljós = light / a light
• ljósið = the light

Ljós is a neuter noun, and ljósið is its nominative singular definite form. In this sentence, ljósið is the subject of the main clause.

Slokknar is the verb form meaning goes out or goes off, used for things like lights, fires, candles, etc.

It comes from the infinitive slokkna. In this sentence, it describes what happens to the light: it goes out.

Slokknar is 3rd person singular present tense of slokkna.

It is singular because the subject is ljósið (the light), which is singular.

A useful pattern to notice is that many Icelandic verbs in -na have a present-tense form in -nar in the 3rd person singular.

This is a very common and important distinction:

• slokkna = to go out, to be extinguished
  
  • intransitive: the thing itself goes out
• slökkva = to turn off, to extinguish
  
  • transitive: someone turns something off

So:

• Ljósið slokknar. = The light goes out.
• Ég slökkvi ljósið. = I turn off the light.

In your sentence, slokknar is used because the light is what undergoes the change.

Because Icelandic often uses the present tense for general truths, repeated situations, and what happens whenever a certain condition occurs.

So this sentence is not just about one single event. It describes a general relationship:

• when a power outage happens, the light goes out

That is very natural in Icelandic, just as it is in English in sentences like Ice melts when it gets warm.

Here, þegar means when.

It introduces a time clause:

• þegar rafmagnsleysi verður = when a power outage occurs

Yes, þegar can also mean already in other contexts.
But in this sentence, it is clearly a conjunction meaning when, because it is followed by a full clause.

Because Icelandic very often forms new nouns by making compounds.

Here:

• rafmagn = electricity
• leysi = lack, absence

So rafmagnsleysi literally means something like lack of electricity, which is why it is used for power outage or power failure.

You may also notice the -s- in the middle: rafmagns-leysi. That linking s is very common in Icelandic compounds.

Verður is the 3rd person singular present tense of verða.

The basic meaning of verða is often become, but in many contexts it can also mean happen or occur.
That is what it does here:

• rafmagnsleysi verður = a power outage occurs / happens

So even if you first learned verða as become, this sentence shows a broader and very common use of the verb.

Because the sentence is talking about a power outage in general, not one specific known outage.

Compare:

• rafmagnsleysi = a power outage / power outage in general
• rafmagnsleysið = the power outage (a specific one)

So the sentence is structured as a general rule, which is why the indefinite form is natural.

It is nominative singular.

That is because it is the subject of verður in the subordinate clause:

• rafmagnsleysi verður

Likewise, ljósið is nominative too, because it is the subject of the main clause:

• Ljósið slokknar

So each clause has its own subject in the nominative.

Then the word order changes because Icelandic follows a verb-second pattern in main clauses.

You get:

• Þegar rafmagnsleysi verður, slokknar ljósið.

Notice that slokknar comes before ljósið in the main clause.
That happens because the whole þegar clause takes the first position, and the finite verb of the main clause must come next.

So:

• Ljósið slokknar þegar rafmagnsleysi verður.
• Þegar rafmagnsleysi verður, slokknar ljósið.

Both are natural, but the second one shows Icelandic word order especially clearly.