Efter morgenmaden tørrer min mor bordet af med en ren klud.

This is because Danish main clauses usually follow the verb-second (V2) rule.

That means:

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

So in this sentence:

• Efter morgenmaden = first element
• tørrer = second element
• min mor = comes after the verb

So the pattern is:

Efter morgenmaden + tørrer + min mor + ...

If the sentence started with the subject instead, you would get:

Min mor tørrer bordet af efter morgenmaden.

Both are correct.

Yes. That is completely natural Danish.

The difference is mainly emphasis:

• Efter morgenmaden tørrer min mor bordet af ... puts the time first, so it highlights when
• Min mor tørrer bordet af efter morgenmaden is more neutral and starts with the subject

English does something similar:

• After breakfast, my mother wipes the table off ...
• My mother wipes the table off after breakfast.

Because Danish often uses the definite form with meals in expressions like after breakfast, before dinner, and so on.

So:

• morgenmad = breakfast
• morgenmaden = the breakfast

In practice, efter morgenmaden is the normal way to say after breakfast when you mean the meal that has just happened.

This is very common with meal words:

• før frokosten = before lunch
• efter aftensmaden = after dinner

So even though English usually says just after breakfast, Danish often prefers the definite form here.

In Danish, the definite article is often attached to the end of the noun instead of being a separate word.

So:

• et bord = a table
• bordet = the table

That is why you do not see a separate word for the here.

This sentence has two definite nouns:

• morgenmaden = the breakfast
• bordet = the table

A separate article like det is usually used when there is an adjective before a definite noun:

• det rene bord = the clean table

But with just the noun by itself, Danish usually adds the definite ending:

• bordet

Here, tørre af is a verb + particle combination, a bit like an English phrasal verb.

It means something like:

• wipe off
• wipe down

So:

• tørrer bordet af = wipes the table off / wipes the table down

The word af is separated because Danish particle verbs are often split in normal sentence structure, especially when there is an object.

So you get:

• tørrer bordet af

not:

• af tørrer bordet
• tørrer af bordet in this meaning

A useful way to think about it is:

• verb first
• object next
• particle after the object

It can mean both, depending on context.

• at tørre can mean to dry
• at tørre af often means to wipe off / wipe down

In this sentence, because the object is bordet and because of af and med en ren klud, the meaning is clearly wipe rather than just dry.

So the context tells you which English word fits best.

Yes. tørrer is the present tense form.

The infinitive is:

• at tørre = to dry / to wipe

The present tense is:

• tørrer = dries / wipes

For many Danish verbs, the present tense is formed by adding -r to the infinitive:

• at tørre → tørrer
• at spise → spiser
• at læse → læser

So min mor tørrer means my mother wipes / is wiping depending on context.

Because the adjective has to agree with the noun.

Here:

• klud is a common gender noun
• so it takes en
• and the adjective in this form is ren

So:

• en ren klud = a clean cloth

Compare:

• et rent håndklæde = a clean towel
  because håndklæde is neuter
• rene klude = clean cloths
  because plural uses -e

So the form ren is correct because klud is an en-word in the singular indefinite.

Med means with here and introduces the tool or means used for the action.

So:

• med en ren klud = with a clean cloth

It tells you how the table is being wiped.

This is a very common use of med in Danish:

• jeg skriver med en blyant = I write with a pencil
• hun skærer brødet med en kniv = she cuts the bread with a knife

Because Danish usually does not put a comma after a short introductory phrase like this.

In English, writers often add a comma after something like After breakfast, ...
In Danish, that comma is normally omitted:

• Efter morgenmaden tørrer min mor bordet af med en ren klud.

Danish commas are used more for clause structure than for short pauses in speech. So a fronted time expression like Efter morgenmaden usually stands without a comma.