Efter lærerens forklaring giver teksten mere mening.

Because Danish often puts a time phrase or other adverbial first for emphasis or flow.

Here, Efter lærerens forklaring means After the teacher’s explanation, and it is placed at the front. In Danish main clauses, when something other than the subject comes first, the verb still has to stay in second position.

So:

• Efter lærerens forklaring = first element
• giver = verb in second position
• teksten = subject after the verb

That is why it is:

• Efter lærerens forklaring giver teksten mere mening.

and not:

• Efter lærerens forklaring teksten giver mere mening.

This is a very common Danish word order pattern.

This is because of the Danish V2 rule (verb-second word order).

In a normal main clause with the subject first, you get:

• Teksten giver mere mening.
  = The text makes more sense.

But when you move another element to the front, such as Efter lærerens forklaring, the verb must come immediately after that fronted element:

• Efter lærerens forklaring giver teksten mere mening.

So the subject teksten moves to a position after the verb.

Here efter means after in a time-related sense: following or once something has happened.

So:

• Efter lærerens forklaring = After the teacher’s explanation

It does not mean according to here. The context is temporal: first the explanation happens, then the text makes more sense.

Lærerens is the genitive form of læreren, which means the teacher.

Step by step:

• lærer = teacher
• læreren = the teacher
• lærerens = the teacher’s

In Danish, possession is often shown by adding -s to the noun:

• manden = the man
• mandens = the man’s

So:

• lærerens forklaring = the teacher’s explanation

Because in Danish, when a noun is possessed, the possessed noun is usually not marked as definite.

So:

• lærerens forklaring = the teacher’s explanation

not:

• lærerens forklaringen

This is similar to English. We say:

• the teacher’s explanation

not:

• the teacher’s the explanation

The possession already makes the phrase specific enough.

Teksten is the normal Danish way to say the text.

Danish usually expresses definiteness by adding an ending to the noun:

• tekst = text
• teksten = the text

Using den tekst is possible, but it usually means something more like that text or that particular text, depending on context. In this sentence, simple definiteness is enough, so teksten is the natural choice.

Literally, giver mening means gives meaning.

But idiomatically, it means:

• makes sense

This is a very common Danish expression:

• Det giver mening. = That makes sense.

So in your sentence:

• teksten giver mere mening
  means
• the text makes more sense

Even though the literal wording looks a bit different from English, the expression works very much like English make sense.

Mere means more, so:

• mere mening = more sense

The sentence is saying that after the teacher’s explanation, the text is easier to understand than before.

This matches English quite closely:

• The text makes more sense now.

You use mere because mening is being treated as something that can increase or decrease in degree here.

Yes, absolutely.

This version is also correct:

• Teksten giver mere mening efter lærerens forklaring.

The difference is mainly one of focus and style.

• Efter lærerens forklaring giver teksten mere mening.
  This emphasizes the time/circumstance first: After the teacher’s explanation...

• Teksten giver mere mening efter lærerens forklaring.
  This starts with the subject and feels a bit more neutral.

Both are natural Danish.

Yes. Forklaring means explanation.

It comes from the verb forklare, which means to explain.

So:

• en forklaring = an explanation
• forklaringen = the explanation
• lærerens forklaring = the teacher’s explanation

Because Efter lærerens forklaring is just a fronted prepositional phrase, not a separate clause.

The sentence has only one finite verb:

• giver

So this is one simple main clause, and no comma is needed.

Yes.

The verb giver is the present tense of give.

So:

• give = to give
• giver = gives / is giving

In this sentence, present tense is used to describe the current result:

• Efter lærerens forklaring giver teksten mere mening.
  = After the teacher’s explanation, the text makes more sense.

A useful way is:

• Efter lærerens forklaring = after the teacher’s explanation
• giver teksten = the text gives / the text makes
• mere mening = more sense

So the full structure is:

• After the teacher’s explanation
  • the text makes
    • more sense

This kind of chunking helps a lot with Danish word order, especially when the sentence starts with something other than the subject.