Kluden ligger ved vasken, så jeg kan tørre bordet af med det samme.

Because -en is the usual definite ending for a common-gender noun in Danish.

• en klud = a cloth
• kluden = the cloth

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

The same pattern appears elsewhere in the sentence:

• vasken = the sink
• bordet = the table

Notice that bordet ends in -et, not -en, because bord is a neuter noun: et bord.

It depends on the noun’s grammatical gender.

In Danish, nouns are usually either:

• common gender: take en in the indefinite form
• neuter: take et in the indefinite form

So:

• en klud → kluden
• en vask → vasken
• et bord → bordet

Unfortunately, this is something you usually have to learn with each noun. It is best to memorize nouns together with en or et, not by themselves.

Danish often uses a position verb where English would simply say is.

Here, ligger literally means lies / is lying. That sounds strange in English, but it is very normal in Danish for things that are resting in a horizontal or loose position.

So:

• Kluden ligger ved vasken = literally The cloth is lying by the sink
• natural English: The cloth is by the sink

Danish commonly uses:

• ligge for things lying or resting flat
• stå for things standing upright
• sidde for things sitting / being attached in a certain way

Ved here means by, near, or next to.

So ved vasken means by the sink or near the sink.

It is used because the cloth is located at the sink area, not necessarily inside it or on top of it.

Compare:

• ved vasken = by the sink
• i vasken = in the sink
• på vasken = on the sink

English learners often want one direct translation for prepositions, but Danish prepositions depend a lot on the exact spatial idea.

Because the sentence has two clauses:

• Kluden ligger ved vasken
• så jeg kan tørre bordet af med det samme

The comma separates them.

Also, så here means so, introducing the result or consequence of the first clause. Danish commonly writes a comma before this kind of clause.

Because så here is functioning as a conjunction meaning so, linking the two clauses together.

When så links clauses like this, the next clause normally begins with the subject:

• så jeg kan tørre bordet af...

If you wrote så kan jeg..., that would sound more like Then I can... or like a new main clause with different emphasis.

So in this sentence:

• så jeg kan... = so I can...

Because kan is a modal verb, and after modal verbs Danish normally uses the infinitive without at.

So:

• jeg kan tørre = I can wipe
• not jeg kan at tørre

This is similar to English:

• I can wipe
• not I can to wipe

Other common Danish modal verbs work the same way:

• jeg vil gå = I want to go / I will go
• jeg skal arbejde = I have to work / I will work
• jeg må vente = I must / may wait

Tørre af is a very common Danish verb combination meaning wipe off, wipe down, or dry off, depending on context.

So:

• tørre bordet af = wipe the table off / wipe down the table

The small word af often adds the idea of removing something from the surface, such as water, dirt, crumbs, or grease.

Without af, tørre can still mean wipe or dry, but tørre af is especially common when talking about cleaning a surface.

Because in Danish, verb particles like af are often separated from the main verb when there is an object.

So you get:

• tørre bordet af
• not usually tørre af bordet in this meaning

This is similar to English phrasal verbs:

• wipe the table off
• turn the light off

The object often comes between the verb and the particle.

You can especially see this clearly with pronouns:

• Jeg tørrer bordet af = I wipe the table off
• Jeg tørrer det af = I wipe it off

It is an idiomatic expression meaning immediately, right away, or at once.

So:

• med det samme = right away

You should learn it as a fixed phrase, because the literal parts do not help much here.

No. In med det samme, det is part of the fixed expression med det samme, meaning right away.

It does not mean with it here.

That is important, because if Danish wanted to say with it, referring back to kluden, it would use den, not det, since klud is a common-gender noun:

• med den = with it, referring to kluden

So:

• med det samme = immediately
• med den = with it

If you wanted to make the cloth explicit, you could say something like:

• så jeg kan tørre bordet af med den med det samme

That means so I can wipe the table with it right away.