Jeg tørrer bordet af med et rent viskestykke, før gæsterne kommer.

Because tørre af is a very common verb combination meaning to wipe off, wipe down, or dry off something.

• at tørre on its own can mean to dry or sometimes to wipe
• at tørre af makes the idea of wiping a surface especially clear

So Jeg tørrer bordet af sounds like I wipe the table down/off, which fits this sentence very well.

This is normal Danish word order for many verb + particle combinations.

With a full noun object, Danish often puts the object before the particle:

• Jeg tørrer bordet af
• Hun skriver navnet ned
• Vi rydder køkkenet op

If the object is a pronoun, the particle still usually comes after it:

• Jeg tørrer det af

So tørrer bordet af is the natural order here.

Because bordet means the table, while bord means a table or just table in a more general sense.

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

• et bord = a table
• bordet = the table

In this sentence, the speaker means a specific table, so bordet is the right form.

Because viskestykke is a neuter noun, so it takes:

• the article et
• the adjective ending -t in the singular indefinite form

So:

• en ren klud
• et rent viskestykke

The adjective ren changes to rent because of the neuter noun.

A quick pattern:

• en ren stol
• et rent bord
• rene stole
• rene borde

A viskestykke is usually a tea towel, dish towel, or kitchen towel.

It is typically a cloth towel used in the kitchen for things like:

• drying dishes
• wiping surfaces
• drying hands

In some contexts, English speakers might also think of dishcloth, but viskestykke often suggests more of a towel than a small rag.

A useful contrast is:

• viskestykke = towel used in the kitchen
• klud = cloth/rag

Because gæsterne means the guests.

The noun forms are:

• en gæst = a guest
• gæster = guests
• gæsterne = the guests

So the sentence is referring to specific guests that both speaker and listener know about.

Because Danish often uses the present tense for future meaning when the context already makes the time clear.

Here, før gæsterne kommer clearly means before the guests come / before the guests arrive.

This is very similar to English, which can also use the present in time clauses:

• before the guests come
• not usually before the guests will come

So kommer is completely normal here.

Because før introduces a subordinate clause.

In Danish subordinate clauses, the usual order is:

conjunction + subject + finite verb

So:

• før gæsterne kommer

not

• før kommer gæsterne

The inverted order kommer gæsterne is more typical of a main clause after something has been placed first.

For example:

• Gæsterne kommer snart
• Snart kommer gæsterne

But after før, the normal subordinate order is gæsterne kommer.

Because før gæsterne kommer is a subordinate clause, and many Danish writers put a comma before subordinate clauses.

So this punctuation is very natural:

• Jeg tørrer bordet af med et rent viskestykke, før gæsterne kommer.

You may also see Danish written without that comma, depending on the comma system being used. Both styles exist, but the version with the comma is very common and easy for learners to recognize.

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

So:

• med et rent viskestykke = with a clean dish towel / tea towel

It tells you how the speaker is wiping the table.

This is very similar to English:

• I wipe the table with a clean cloth
• Jeg tørrer bordet af med et rent viskestykke