Síðan við fluttum hingað þekkjum við fleiri íbúa í húsinu.

Here síðan means since, introducing a time clause:

• Síðan við fluttum hingað = Since we moved here

Be careful: síðan can also mean then / afterwards in other contexts, but in this sentence it is a conjunction meaning since.

That is very natural here.

• við fluttum hingað = we moved here → a completed event in the past
• við þekkjum fleiri íbúa = we know more residents → the present result or current situation

So the sentence means something like:

• Since we moved here, we know more residents in the building.

This is similar to English, where the since-clause often refers to a past event, while the main clause describes the present.

Fluttum is the 1st person plural past tense of flytja (to move).

So:

• ég flutti = I moved
• við fluttum = we moved

Here it matches við (we).

Because hingað expresses movement toward a place, while hér expresses location.

• hingað = to here
• hér = here

So:

• við fluttum hingað = we moved here
• not we moved here/stayed here in the sense of location

This is a very common Icelandic distinction:

• hér = here
• hingað = to here
• héðan = from here

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

The sentence begins with the subordinate clause:

• Síðan við fluttum hingað

After that whole clause, the main clause begins, and its finite verb must come in the second position:

• þekkjum við fleiri íbúa í húsinu

So the order is not random. It is a normal Icelandic pattern:

• [fronted element] + [finite verb] + [subject]

Compare:

• Við þekkjum fleiri íbúa í húsinu.
• Síðan við fluttum hingað þekkjum við fleiri íbúa í húsinu.

Þekkja means to know in the sense of being familiar with a person, recognizing, or knowing someone/something personally.

So:

• við þekkjum fleiri íbúa = we know more residents

This is the right verb for knowing people.

It is different from vita, which is used for knowing a fact or being aware of something.

Very roughly:

• þekkja mann = know a person
• vita eitthvað = know something / know that...

Fleiri means more when talking about countable plural nouns.

Here it goes with íbúa (residents), which is plural/countable:

• fleiri íbúa = more residents

This is the correct comparative form here.

Íbúa is in the accusative plural because þekkja normally takes a direct object in the accusative.

So:

• að þekkja einhvern = to know someone
• við þekkjum íbúa = we know residents

The noun is íbúi (resident), and here the plural object form is íbúa.

Because Icelandic uses the preposition í for in.

• í húsinu = in the building

So the phrase means that these residents are residents in the building.

Without í, the meaning would not work the same way.

Húsinu is the dative singular definite form of hús (house/building).

That happens because:

1. the preposition í often takes the dative when it means location
2. the noun is definite: the building

So:

• í húsi = in a building
• í húsinu = in the building

This is a very important Icelandic pattern:

• í + dative for location
• í + accusative for movement into something

Here the meaning is location, so dative is used.

It can mean either, depending on context.

• hús often means house
• but it can also mean building

In this sentence, because of íbúa í húsinu (residents in the building), building is probably the most natural English translation.

Possibly, depending on context, but more residents is the most direct translation of fleiri íbúa.

Usually fleiri is simply comparative:

• more

So the basic idea is that after moving here, they now know a greater number of residents than before.

They are very close in meaning here.

• síðan við fluttum hingað = since we moved here
• frá því að við fluttum hingað = since we moved here

Frá því að can sound a bit more explicit, while síðan is shorter and very common. In many contexts, either would work.