Während meine Mutter die Paprika schneidet, bereite ich schon den Quark für morgen vor.

Während means while here. It introduces a subordinate clause, and in German that sends the conjugated verb to the end of that clause.

So:

• Während meine Mutter die Paprika schneidet
• literally: While my mother the peppers cuts

That is why schneidet comes at the end of the first clause.

In German, a subordinate clause is normally separated from the main clause by a comma. Since Während meine Mutter die Paprika schneidet is a subordinate clause, it must be followed by a comma before the main clause begins:

• Während meine Mutter die Paprika schneidet, bereite ich schon den Quark für morgen vor.

This comma is not optional in standard German.

German main clauses usually follow the verb-second rule. The conjugated verb must be in the second position.

In this sentence, the whole subordinate clause comes first:

• Während meine Mutter die Paprika schneidet

That first clause counts as position 1. So in the main clause, the conjugated verb must come immediately after it:

• bereite ich schon den Quark für morgen vor

So bereite ich is correct.

If you started with the main clause instead, you would say:

• Ich bereite schon den Quark für morgen vor, während meine Mutter die Paprika schneidet.

Because vorbereiten is a separable verb.

Its base form is:

• vorbereiten = to prepare

In a normal main clause, separable verbs split apart:

• ich bereite ... vor

So in the sentence:

• bereite ich schon den Quark für morgen vor

the bereite part is the conjugated verb, and vor goes to the end of the clause.

Because Quark is the direct object of vorbereiten, so it is in the accusative case.

• nominative: der Quark
• accusative: den Quark

Here, the subject is ich, and the thing being prepared is den Quark.

Here die Paprika is most naturally understood as the peppers. In German, Paprika can refer to peppers, and the form can look the same in singular and plural depending on context.

In this sentence, because someone is cutting them for food preparation, learners often understand it as plural: the peppers.

So:

• die Paprika schneidet = is cutting the peppers

Even though die can also be a singular feminine article, context often helps you decide what is meant.

Because meine Mutter is the subject of the subordinate clause, so it is in the nominative case.

In:

• Während meine Mutter die Paprika schneidet

the mother is the one doing the cutting. That makes her the subject.

So:

• meine Mutter = nominative
• meiner Mutter would be dative or genitive in other contexts, which is not needed here

Schon here means something like already.

• bereite ich schon den Quark ... vor
• I’m already preparing the quark ...

It suggests that the speaker is getting started early or is ahead in the process.

Be careful: schon can have different meanings in different contexts, such as indeed, quite, or all right, but here already is the natural meaning.

German often uses the present tense to talk about the future when the time is clear from context.

So:

• bereite ich ... für morgen vor

literally uses present tense, but it means something like:

• I’m preparing ... for tomorrow

This is very normal in German. You do not need a special future tense just because it refers to tomorrow.

Für morgen means for tomorrow and tells us when/for what time the quark is being prepared.

Its placement here is natural:

• bereite ich schon den Quark für morgen vor

But German word order is flexible, so other versions are possible, for example:

• Während meine Mutter die Paprika schneidet, bereite ich den Quark schon für morgen vor.
• Während meine Mutter die Paprika schneidet, bereite ich für morgen schon den Quark vor.

Some versions sound more natural than others, but they are grammatically possible. The original sentence is smooth and idiomatic.

That is a very common learner question because während can be both a preposition and a subordinating conjunction.

In this sentence, während is a conjunction, so it introduces a clause:

• während meine Mutter die Paprika schneidet

As a conjunction, it does not govern a noun case directly.

But während can also be a preposition, and then it often takes the genitive in formal German:

• während des Essens = during the meal

So in your sentence, it is not a case issue; it is a clause-introducing word meaning while.