Í dag biðum við fimmtán mínútur eftir strætónum og gengum síðan sextán mínútur heim.

Because Icelandic main clauses usually follow the verb-second rule.

That means:

• the first position can be a time phrase, place phrase, object, etc.
• the finite verb still comes second

So:

• Við biðum í dag ... = neutral order
• Í dag biðum við ... = Í dag is moved to the front for emphasis/topic, so biðum must come next

This is one of the most important word-order patterns in Icelandic.

Biðum is the 1st person plural past tense of bíða.

So:

• bíða = to wait
• beið = waited, singular past
• biðum = waited, plural past

In this sentence, biðum við means we waited.

Because Icelandic normally does not drop subject pronouns the way some languages do.

Even though -um helps show that the verb is plural, Icelandic usually still says the pronoun:

• við biðum = we waited

So the verb ending gives useful grammatical information, but it usually does not replace the subject pronoun in an ordinary sentence.

Because the subject is the same as in the first clause, and Icelandic often leaves out a repeated subject in coordinated clauses.

So:

• Í dag biðum við ... og gengum síðan ...

means:

• Today we waited ... and [we] then walked ...

You could repeat við, but it is often unnecessary:

• ... og við gengum síðan ...

That version is possible, but the sentence without the repeated pronoun is very natural.

On its own, eftir often does mean after, but with bíða it forms the expression bíða eftir, which means wait for.

So:

• eftir = after / following, in many contexts
• bíða eftir = to wait for

This is something you simply learn as a verb + preposition combination.

Because eftir takes the dative, and strætónum is the dative singular definite form of strætó.

Breakdown:

• strætó = bus
• strætónum = the bus in the dative singular

The ending -num includes the definite article and the dative ending.

So after bíða eftir, you get:

• eftir strætónum = for the bus

They are functioning as accusative expressions of duration.

Icelandic often uses a bare time expression, without a preposition, to show how long something lasted:

• biðum fimmtán mínútur = waited fifteen minutes
• gengum sextán mínútur = walked sixteen minutes

With mínúta, the plural form mínútur looks the same in the nominative and accusative, so the form itself does not change here.

Because Icelandic often expresses duration directly, without a word like for.

English says:

• waited for fifteen minutes
• walked for sixteen minutes

Icelandic very naturally says:

• biðum fimmtán mínútur
• gengum sextán mínútur

So the idea of for is built into the time expression rather than being shown by a separate preposition.

Because heim shows motion toward home, while heima shows location at home.

Compare:

• fara heim = go home
• vera heima = be at home

Since the sentence describes movement, gengum ... heim, the directional form heim is the correct one.

Síðan means then, after that, or afterwards.

It shows the sequence of events:

1. we waited for the bus
2. then we walked home

So it works as a time adverb linking the two actions in order.

Because ganga is an irregular strong verb.

Its past tense changes the stem vowel:

• ganga = to walk
• gekk = walked, singular past
• gengum = walked, plural past

So this is not a regular pattern you can build mechanically from the infinitive; it is one of the verb forms you need to learn.

Yes. That would also be grammatical.

But when the same subject continues into the second clause, Icelandic often prefers to leave the repeated subject out:

• ... og gengum síðan heim

If you do include við, it can sound slightly more explicit or slightly more emphatic:

• ... og við gengum síðan heim

Both are possible, but the version without the second við is very natural.