Ich nehme ein Stück Küchenrolle, um den Fleck auf dem Tisch abzuwischen.

Eine Küchenrolle usually means a whole roll of kitchen paper / paper towels.

Ein Stück Küchenrolle means a piece of kitchen paper, so just one sheet or torn-off piece. German often uses Stück to talk about one piece of something rather than the whole item.

So:

• eine Küchenrolle = a whole roll
• ein Stück Küchenrolle = a piece of paper towel

Küchenrolle literally means kitchen roll. In natural English, this is usually paper towel or a roll of paper towels, depending on context.

It is made from:

• Küche = kitchen
• Rolle = roll

In this sentence, because of ein Stück, it refers to a piece torn from a paper towel roll.

Because abwischen takes a direct object in the accusative case.

The noun is:

• der Fleck = the stain, the spot

But as the direct object, it changes:

• der Fleck → den Fleck

So here:

• Ich wische den Fleck ab = I wipe away the stain

This is a very common pattern in German: masculine nouns change from der to den in the accusative.

Because this phrase describes a location, not a movement toward somewhere.

German two-way prepositions like auf use:

• dative for location: where?
• accusative for direction/movement: where to?

Here the stain is already located on the table, so German uses the dative:

• auf dem Tisch = on the table

Compare:

• Der Fleck ist auf dem Tisch. = The stain is on the table.
• Ich lege das Buch auf den Tisch. = I put the book onto the table.

So in your sentence, dem Tisch is correct because it answers where?

Tisch is a masculine noun:

• der Tisch

After auf in a location meaning, you need the dative. The dative form of der is dem:

• der Tisch → dem Tisch

So:

• auf dem Tisch = on the table

Um ... zu is used to express purpose: in order to, or more naturally in English, often just to.

So:

• Ich nehme ein Stück Küchenrolle, um den Fleck auf dem Tisch abzuwischen.
• I take a piece of paper towel in order to wipe the stain off the table.

Structure:

• um
  • other parts of the clause + zu + infinitive

This construction is used when the subject of both actions is the same. Here, I am both taking the paper towel and wiping the stain.

Because abwischen is being used in an um ... zu clause.

With zu, a separable verb does not split in the normal way. Instead, zu goes between the prefix and the base verb:

• abwischen → abzuwischen

Compare:

• Ich wische den Fleck ab.
  Here the verb is conjugated, so it splits.

• ..., um den Fleck abzuwischen.
  Here it is an infinitive with zu, so it becomes one word: abzuwischen.

This is normal for separable verbs:

• aufmachen → aufzumachen
• mitnehmen → mitzunehmen
• abwischen → abzuwischen

Wischen means to wipe in a general sense.

Abwischen often means to wipe off, to wipe away, or to wipe clean. The prefix ab- adds the idea of removing something from a surface.

So here abwischen is a good choice because the goal is to remove the stain from the table.

Very roughly:

• wischen = wipe
• abwischen = wipe off / wipe away

German often uses the present tense for actions happening now, very soon, or in a normal sequence of actions.

So Ich nehme ... can mean:

• I take ...
• I’m taking ...
• I’m going to take ...

The exact English translation depends on context, but German does not need a separate future form here.

This is a common partitive-style expression in German.

When you say ein Stück Küchenrolle, the second noun often appears without an article, because it works a bit like a material or substance:

• ein Stück Kuchen = a piece of cake
• ein Glas Wasser = a glass of water
• eine Tasse Kaffee = a cup of coffee
• ein Stück Küchenrolle = a piece of paper towel

So Küchenrolle here is not the main noun of the phrase. The main noun is Stück, and Küchenrolle tells you what kind of piece it is.

The sentence has two parts:

1. Main clause: Ich nehme ein Stück Küchenrolle
2. Purpose clause: um den Fleck auf dem Tisch abzuwischen

In the main clause, the normal German pattern is:

• subject
  • finite verb
    • other elements

So:

• Ich = subject
• nehme = finite verb
• ein Stück Küchenrolle = object

In the um ... zu clause, the infinitive goes at the end:

• um den Fleck auf dem Tisch abzuwischen

That final verb position is completely normal in German subordinate-style structures.

Yes, that can also sound natural, but it changes the nuance slightly.

• den Fleck auf dem Tisch abwischen focuses on the stain located on the table
• den Fleck vom Tisch abwischen emphasizes wiping the stain off the table

In many contexts, both are possible. The version with vom Tisch makes the idea of removal a bit more explicit.

So:

• auf dem Tisch = on the table
• vom Tisch = off/from the table

Both can work, depending on what exactly you want to highlight.