Jeg tørrer skærebrættet af med en ren klud.

Tørre is the infinitive, the form you would use after at: at tørre.

Tørrer is the present tense:

• jeg tørrer = I am drying / I dry
• du tørrer = you are drying / you dry

So in a normal present-tense sentence with jeg, Danish needs tørrer.

Because tørre ... af is a verb + particle combination, similar to English verbs like wipe off.

The basic expression is at tørre noget af. In a main clause, the finite verb comes early, but the particle af stays later in the sentence:

• Jeg tørrer skærebrættet af
• Hun tørrer bordet af

So this is normal Danish word order. You should think of tørre af as belonging together in meaning, even though the two parts are separated.

Both can relate to making something dry, but tørre ... af often has the sense of wiping something off / wiping it dry, especially with a cloth.

So:

• tørre noget = dry something
• tørre noget af = wipe something off, or wipe it dry

In this sentence, because there is med en ren klud, tørre ... af sounds especially natural.

Because skærebrættet is the definite form: the cutting board.

Danish usually adds the definite article to the end of the noun instead of using a separate word like English the.

• et skærebræt = a cutting board
• skærebrættet = the cutting board

So the sentence is talking about a specific cutting board, not just any cutting board.

The base noun is et skærebræt.

It is a neuter noun, so its definite ending is -et. When the definite form is added, you get:

• skærebræt
  • -et → skærebrættet

The spelling with tt is just part of the normal written form here. The important thing for a learner is:

• et skærebræt = indefinite
• skærebrættet = definite

Yes. Danish makes compound nouns very freely.

Skærebræt is made from:

• skære = cut
• bræt = board

So it is literally something like cutting-board.

This is very common in Danish, and English speakers usually benefit from learning to spot these compounds. Once you recognize the parts, long Danish nouns become much easier to understand.

Because klud is a common gender noun, so it takes en, not et:

• en klud = a cloth

And with a common gender singular indefinite noun, the adjective usually has its basic form:

• en ren klud

Compare:

• en ren klud = a clean cloth
• et rent håndklæde = a clean towel

So rent would be used with a neuter noun, but klud is not neuter.

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

So med en ren klud tells you what the speaker is using to wipe the cutting board.

This is very common in Danish:

• Jeg skriver med en blyant = I write with a pencil
• Hun åbner døren med en nøgle = She opens the door with a key

In a normal main clause, Danish keeps the finite verb in second position.

The basic sentence is:

• Jeg tørrer skærebrættet af med en ren klud

If you move another element to the front for emphasis, the verb still stays second:

• Med en ren klud tørrer jeg skærebrættet af

That is a very important Danish pattern.

Also, the particle af normally stays with the verbal expression and comes after the object:

• Jeg tørrer det af
• Jeg tørrer skærebrættet af

So you should not put af directly before tørrer in this sentence.

Then the pronoun usually comes before af:

• Jeg tørrer det af
• Hun tørrer den af

This is the same pattern you see with the full noun:

• Jeg tørrer skærebrættet af

So whether the object is a full noun or a pronoun, af still belongs to the verb phrase, but the object comes before it.

They are both vowels that English speakers often need to practice.

• ø in tørrer has no exact English equivalent. A useful approximation is: say something like eh, but with rounded lips.
• æ in skærebræt is closer to the vowel in cat, though Danish pronunciation can vary by word and speaker.

A few rough guides:

• tørrer ≈ something like TUR-er, but with a Danish ø
• skærebræt ≈ something like SKEH-re-bret
• klud has a long u sound, roughly like klood, though the Danish vowel is not exactly the same as English oo

These are only approximations, but they can help you get started.