Efter at vi har spist aftensmad, slapper vi af i stuen.

Efter at means after when it introduces a whole clause.

• efter = after
• at = that / to, but here it helps introduce the clause

So Efter at vi har spist aftensmad literally means After that we have eaten dinner, but in natural English: After we have eaten dinner or simply After we eat dinner.

Har spist is the present perfect:

• har = have
• spist = eaten

So vi har spist = we have eaten.

In this sentence, Danish uses the present perfect in the time clause to show that the eating is completed before the relaxing happens. English often does the same:

• After we have eaten dinner, we relax...

But in everyday English, we often simplify it to:

• After we eat dinner, we relax...

Danish commonly keeps har spist in this kind of sentence.

This is because Danish has verb-second word order in main clauses.

When the sentence starts with something other than the subject, the finite verb comes before the subject.

Here, the sentence begins with the subordinate clause:

• Efter at vi har spist aftensmad

So in the main clause, Danish puts the verb first:

• slapper = finite verb
• vi = subject

That gives:

• Efter at vi har spist aftensmad, slapper vi af i stuen.

If the sentence began directly with the subject, it would be:

• Vi slapper af i stuen efter at vi har spist aftensmad.

Both are correct.

Slapper af means relax or unwind.

It comes from the verb at slappe af. This is a phrasal / particle verb, where the meaning belongs to the combination, not just the first word alone.

• slapper = relaxes / relax
• af = off, but here it is just part of the verb

So you should learn it as one unit:

• at slappe af = to relax

In main clauses, the particle often stays later in the sentence:

• Vi slapper af
• Han slapper altid af efter arbejde

In an infinitive, it stays together:

• Vi vil gerne slappe af

Aftensmad means evening meal / dinner.

This can be confusing for English speakers because meal words do not always match perfectly across languages.

In modern Danish:

• morgenmad = breakfast
• frokost = lunch
• aftensmad = dinner / evening meal

You may also hear middag in some contexts, but aftensmad is very common for the evening meal in everyday Danish.

So in this sentence, aftensmad is simply dinner.

I stuen means in the living room.

• i = in
• stuen = the living room / lounge

The ending -en is the definite article attached to the noun:

• stue = living room
• stuen = the living room

So Danish often puts the at the end of the noun instead of using a separate word.

The comma separates the introductory subordinate clause from the main clause.

• Efter at vi har spist aftensmad = subordinate clause
• slapper vi af i stuen = main clause

In Danish, commas are commonly used to mark this boundary. So the comma helps show where the first clause ends and the main statement begins.

Yes, that is grammatically correct.

It means the same thing, but the focus is slightly different:

• Efter at vi har spist aftensmad, slapper vi af i stuen.
  This emphasizes the time sequence first.

• Vi slapper af i stuen, efter at vi har spist aftensmad.
  This starts with the main action and adds the time information afterward.

In everyday speech, both are natural.

Danish does not need an extra word like then here.

The sequence is already clear from:

• Efter at = after
• the completed action har spist
• the following main clause slapper vi af

English sometimes says:

• After dinner, then we relax...

But then is optional there too, and in Danish it would usually be unnecessary.

Sometimes you may hear efter vi... in informal speech, but efter at vi... is the standard and safest form for learners.

So for correct written Danish and careful spoken Danish, use:

• Efter at vi har spist aftensmad...

That is the form you should learn first.

A rough guide for an English speaker:

• slapper af: something like SLA-bor af
  The exact sounds are softer and more reduced than English spelling suggests.

• stuen: roughly STOO-en or STU-en, but with a very light middle sound

A few pronunciation points:

• Danish a in slapper is not exactly the English a in slap
• final sounds are often softer than learners expect
• unstressed syllables get reduced a lot in natural speech

If you want, I can also give you the whole sentence in a simple pronunciation guide for English speakers.