Stigagangurinn er sameiginlegur, svo allir íbúar þurfa að halda honum hreinum.

Stigagangurinn means the stairwell / the staircase area / the shared hallway around the stairs in a building.

It is made up of:

• stigagangur = stairwell, staircase hall
• -inn = the attached definite article, meaning the

So:

• stigagangur = a stairwell
• stigagangurinn = the stairwell

This kind of attached the is very common in Icelandic.

Because sameiginlegur has to agree with stigagangurinn.

Here:

• stigagangurinn is masculine
• singular
• nominative (because it is the subject)

So the adjective must also be:

• masculine
• singular
• nominative

That gives sameiginlegur.

Compare:

• sameiginlegur = masculine singular nominative
• sameiginlegt = neuter singular nominative/accusative
• sameiginleg = feminine singular nominative

After er (is), Icelandic adjectives still agree with the noun they describe.

Here svo means so, therefore, or as a result.

So the logic is:

• The stairwell is shared
• so all residents need to keep it clean

In other contexts, svo can also mean things like so, then, or thus, depending on the sentence. But in this sentence, the natural meaning is clearly therefore / so.

Allir íbúar means all residents.

Breakdown:

• allir = all, masculine plural nominative
• íbúar = residents, masculine plural nominative

This is the subject of the second clause, so it has to be in the nominative plural.

It is also indefinite here:

• allir íbúar = all residents / all the people living there, in general
• allir íbúarnir = all the residents (more explicitly definite)

Both can be possible in Icelandic depending on style and context, but allir íbúar is very natural for a general statement.

Because the subject is plural: allir íbúar.

The verb að þurfa means to need / to have to. In the present tense:

• ég þarf = I need
• þú þarft = you need
• hann/hún/það þarf = he/she/it needs
• við þurfum = we need
• þið þurfið = you need
• þeir/þær/þau þurfa = they need

Since allir íbúar is a plural subject, Icelandic uses þurfa.

Because þurfa is followed by the infinitive with að in Icelandic.

So:

• þurfa að halda = need to keep
• literally, something like need to keep

This is different from English in some places, because English sometimes uses a bare infinitive after modal verbs, but Icelandic often keeps að before the infinitive.

So the structure is:

• allir íbúar þurfa að halda...
• all residents need to keep...

Honum refers back to stigagangurinn.

So it means:

• honum = it, referring to the stairwell

But grammatically, it is the dative singular masculine form of the pronoun.

That may look strange to an English speaker, because English just says it, but Icelandic changes the pronoun form depending on grammatical case.

The masculine singular forms are:

• hann = nominative
• hann = accusative
• honum = dative
• hans = genitive

Here the verb pattern is halda einhverju hreinu = keep something clean, and in this pattern the thing being kept clean is in the dative, so Icelandic uses honum.

Because hreinum agrees with honum.

In halda honum hreinum:

• honum is masculine singular dative
• so the adjective describing it must also be masculine singular dative
• that form is hreinum

So the pattern is:

• halda einhverju hreinu = keep something clean

In this sentence, because the noun referred to is masculine:

• honum hreinum = it clean

If the noun were neuter, you would get a different form. So this is another example of Icelandic adjectives agreeing in gender, number, and case.

This is a very useful Icelandic structure:

• að halda einhverju hreinu = to keep something clean

More generally:

• að halda einhverju + adjective

Examples:

• halda herberginu hreinu = keep the room clean
• halda gluggunum opnum = keep the windows open
• halda matnum heitum = keep the food warm

So in your sentence:

• halda = keep
• honum = it, the stairwell
• hreinum = clean

This is not exactly the same as English word-for-word, but the whole structure functions very much like English keep something clean.

Yes. Icelandic could repeat the noun, but the pronoun is more natural here.

You could theoretically say:

• ... þurfa að halda stigaganginum hreinum

That would mean:

• ... need to keep the stairwell clean

Notice that if you repeat the noun, it changes from stigagangurinn to stigaganginum, because now it is in the dative, just like honum.

So the sentence shows two different case forms of the same noun:

• stigagangurinn = nominative, as subject
• honum / stigaganginum = dative, after halda in this pattern

Using honum avoids repetition and sounds very natural.