Ich wische die Krümel mit der Küchenrolle weg und hoffe, dass der Kratzer im Holz nicht größer wird.

Because wegwischen is a separable verb.

• The full infinitive is wegwischen = to wipe away
• In a normal main clause, the verb splits:
  • Ich wische die Krümel weg.
• The conjugated part (wische) stays in the usual verb position, and the prefix (weg) goes to the end.

Compare:

• Ich wische die Krümel weg.
• Ich will die Krümel wegwischen.
  Here it does not split, because the infinitive stays together.

Krümel means crumbs here, and it is plural.

A few useful points:

• singular: der Krümel = the crumb
• plural: die Krümel = the crumbs

In this sentence, die Krümel is the direct object of wischen, so it is in the accusative case. But for this noun, the plural accusative looks exactly like the plural nominative:

• nominative plural: die Krümel
• accusative plural: die Krümel

So the form does not change.

Because the preposition mit always takes the dative case.

• die Küchenrolle = nominative
• mit der Küchenrolle = dative

So after mit, feminine die becomes der.

This is a very common pattern:

• mit der Gabel = with the fork
• mit der Hand = with the hand
• mit der Küchenrolle = with the kitchen roll / paper towel

Küchenrolle usually means kitchen roll or paper towel roll.

Depending on the variety of English:

• British English: kitchen roll
• American English: paper towel / paper towel roll

So mit der Küchenrolle means something like with the paper towel or using the kitchen roll.

Because dass introduces a subordinate clause, and in German subordinate clauses, the conjugated verb goes to the end.

Main clause word order:

• Ich hoffe.

Subordinate clause with dass:

• dass der Kratzer im Holz nicht größer wird

So:

• Ich hoffe, dass ...
• ..., dass der Kratzer im Holz nicht größer wird.

This is one of the most important German word order rules.

Compare:

• Der Kratzer wird nicht größer.
  main clause
• ..., dass der Kratzer nicht größer wird.
  subordinate clause

größer wird means gets bigger or becomes bigger, not just is bigger.

So the sentence expresses a change:

• nicht größer wird = does not get bigger

If you said:

• ..., dass der Kratzer nicht größer ist

that would mean:

• ..., that the scratch is not bigger

That sounds more like a comparison with something else, not a fear that it might grow or worsen.

So wird is the natural choice here.

Here, nicht is negating the idea of becoming bigger.

• größer wird = gets bigger
• nicht größer wird = does not get bigger

German often places nicht directly before the part being negated, especially in phrases like this.

So:

• ..., dass der Kratzer im Holz nicht größer wird
  = ..., that the scratch in the wood doesn’t get bigger

This placement is very natural and common.

Because der Kratzer is the subject of the subordinate clause.

In:

• ..., dass der Kratzer im Holz nicht größer wird

the thing that becomes bigger is the scratch, so der Kratzer is nominative.

A quick breakdown:

• der Kratzer = subject
• im Holz = location
• wird = verb

If it were a direct object, then you might expect den Kratzer, but here it is not an object.

im Holz is short for in dem Holz.

Literally, it means in the wood.

In this sentence, that suggests the scratch is in the material itself, not just sitting on top of it. That is why im Holz makes sense.

Compare:

• ein Kratzer im Holz = a scratch in the wood
• ein Fleck auf dem Holz = a stain on the wood

So German is focusing on the scratch as damage in the wooden surface/material.

Because the subject Ich is understood for both verbs:

• Ich wische ... weg und hoffe ...

In English, we often do the same:

• I wipe away the crumbs and hope ...

You could repeat ich in some contexts, but here it would be unnecessary and unnatural.

So the structure is:

• Ich wische ... weg
• und [ich] hoffe ...

The second ich is simply omitted because it is the same subject.

Yes. Depending on the exact nuance, you might also hear:

• abwischen = wipe off / wipe away
• aufwischen = mop up / wipe up
• wegmachen = remove, get rid of

But wegwischen is a very good choice when the idea is specifically wiping something away so that it is gone.

So:

• Ich wische die Krümel weg sounds natural and clear.

It highlights the result: the crumbs are removed by wiping.