Eftir síðasta lagið dönsum við aðeins á dansgólfinu áður en við förum út úr salnum.

Because Icelandic is a verb-second (V2) language in main clauses.

The first element in the sentence is Eftir síðasta lagið. Once that time phrase takes first position, the finite verb must come second:

• Eftir síðasta lagið dönsum við ...

If there were no fronted time phrase, the more basic order would be:

• Við dönsum aðeins á dansgólfinu ...

So dönsum við is not strange word order; it is normal Icelandic after something else has been moved to the front.

Here eftir means after, and in this time expression it takes the accusative.

So síðasta lagið is accusative singular:

• síðasta = accusative singular weak form, neuter
• lagið = accusative singular definite of lag (song)

Because lag is neuter, the nominative and accusative singular look the same, so you cannot see the case difference on lagið itself. But grammatically, this phrase is accusative.

Because it agrees with lagið.

The noun lag is:

• neuter
• singular
• definite (the song)

When an adjective modifies a definite noun in Icelandic, it usually takes the weak form. For neuter singular here, that gives síðasta.

So:

• síðasta lagið = the last song

Compare other possible forms:

• síðasti maðurinn = the last man
• síðasta konan = the last woman
• síðustu lögin = the last songs

Aðeins often means only, just, or a little / for a bit, depending on context.

In this sentence, it most naturally suggests something like:

• just
• for a little while
• a bit

So dönsum við aðeins gives the idea that we dance a little / just dance briefly before leaving.

Because á can take different cases depending on meaning:

• dative for location: on / in a place
• accusative for motion toward a place

Here the meaning is we dance on the dance floor, so it is a location, not movement onto it. That is why Icelandic uses the dative:

• á dansgólfinu = on the dance floor

If the idea were onto the dance floor, you would expect accusative instead:

• á dansgólfið

Áður en means before and introduces a subordinate clause.

So:

• áður en við förum út úr salnum
• before we go out of the hall

After áður en, the clause has normal subordinate-clause order:

• við förum
• not förum við

That is different from the main clause, where Icelandic uses verb-second order.

Förum is the 1st person plural present tense of fara (to go).

So literally:

• við förum = we go

In both Icelandic and English, the present tense is often used after time expressions like before, after, when, especially for planned or expected future actions.

So áður en við förum is perfectly natural and means:

• before we go

Because they do different jobs:

• út = out (directional adverb)
• úr = out of / from (preposition)

Together, út úr means out of in a very natural Icelandic way.

Compare:

• Við förum út. = We go out.
• Við förum úr salnum. = We leave the hall / go from the hall.
• Við förum út úr salnum. = We go out of the hall.

So it is not unnecessary repetition; it is a normal combination.

Because úr takes the dative case.

The base form is:

• salur = hall

Its definite dative singular form is:

• salnum = the hall in the dative

So:

• úr salnum = out of the hall

Because Icelandic usually expresses the definite article the as a suffix attached to the noun, not as a separate word.

So:

• lagið = the song
• dansgólfinu = the dance floor (dative)
• salnum = the hall (dative)

The exact ending changes depending on:

• gender
• number
• case

That is why the attached the does not always look the same.

Because Icelandic normally states the subject in each finite clause.

So the sentence has:

• main clause: dönsum við
• subordinate clause: áður en við förum

Even though the subject is the same in both clauses, Icelandic normally keeps við rather than leaving it out. That makes the sentence clear and grammatically complete.