De tager afsted efter grillaftenen.

It can mean either, depending on context.

At the start of a sentence, de is capitalized anyway, so De does not automatically tell you which meaning is intended.

• de = they
• De = formal, polite you

In modern everyday Danish, de meaning they is much more common. The polite De still exists, but it sounds formal or old-fashioned in many situations.

Tage afsted is an idiomatic expression meaning to leave, to head off, or to set off.

So De tager afsted means something like:

• They leave
• They head off
• You leave if the formal De is intended

It is not usually understood literally as take away here. The combination tage + afsted works as a set expression.

Here afsted means away / off / on one’s way.

In this kind of expression, Danish very often writes it as afsted. You may also come across af sted in some texts, but learners should recognize that both refer to the same basic idea of movement away from a place.

In tage afsted, it helps give the sense of going off or setting out.

Efter means after.

It introduces a time expression: efter grillaftenen = after the barbecue evening.

So the sentence tells you when the leaving happens: it happens after that event.

Because Danish is using the definite form.

• grillaften = a barbecue evening / barbecue night
• grillaftenen = the barbecue evening / that barbecue evening

The ending -en is the definite article attached to the noun. English uses a separate word, the, but Danish often adds it to the end of the noun.

So efter grillaftenen means after the barbecue evening, referring to a specific one.

Yes. Danish makes compound nouns very freely.

grillaftenen can be broken down like this:

• grill = grill / barbecue
• aften = evening
• -en = the definite ending

So:

• grillaften = barbecue evening
• grillaftenen = the barbecue evening

This is very typical Danish word-building.

This is normal Danish main-clause word order:

• De = subject
• tager = finite verb
• afsted = adverb/particle-like element connected to the verb
• efter grillaftenen = time expression

Danish main clauses normally put the finite verb in the second position. Since the subject comes first here, the verb comes right after it.

It is also natural to keep tager afsted together, because it functions as a fixed expression. The time phrase then comes afterward.

You would normally say:

De tager ikke afsted efter grillaftenen.

So ikke goes after the finite verb tager in this main clause.

That gives the meaning:

• They are not leaving after the barbecue evening
• or You are not leaving after the barbecue evening, if De is formal you

Yes. Danish often allows that.

You can say:

Efter grillaftenen tager de afsted.

That still means the same basic thing, but it puts more focus on the time expression after the barbecue evening.

Notice that when the time phrase comes first, the verb still stays in second position:

• Efter grillaftenen = first position
• tager = second position
• de = after the verb

That is a very typical Danish pattern.