Für das Abendessen schneide ich eine Aubergine und zwei Süßkartoffeln in kleine Stücke.

Because Für das Abendessen has been put at the beginning of the sentence.

In a normal German main clause, the finite verb must be in the second position. So when a different element comes first, the subject usually moves after the verb.

• Ich schneide eine Aubergine ... = neutral word order
• Für das Abendessen schneide ich eine Aubergine ... = Für das Abendessen takes the first slot, so schneide stays second, and ich comes after it

This is the standard verb-second (V2) pattern in German.

Für always takes the accusative case.

So:

• das Abendessen is accusative here because it follows für

Also, Abendessen is a neuter noun, so its article is das. In neuter singular, the nominative and accusative forms are both das, so the form does not change.

Yes. German very often combines words into a single compound noun.

Here:

• Abend = evening
• Essen = food / meal / eating

Together:

• Abendessen = evening meal / dinner

And because it is a noun, it is capitalized: Abendessen.

This is very common in German:

• Frühstück
• Mittagessen
• Abendessen

Because Aubergine is singular, while Süßkartoffeln is plural.

• eine Aubergine = one aubergine / eggplant
• zwei Süßkartoffeln = two sweet potatoes

German has no indefinite article in the plural, just like English does not say a two sweet potatoes. The number word zwei already tells you how many.

Also, Aubergine is a feminine noun, so its singular indefinite article is eine.

The singular is:

• die Süßkartoffel

The plural is:

• die Süßkartoffeln

Many feminine German nouns form the plural with -n or -en. Since Süßkartoffel already ends in -el, the plural is formed with -n:

• Kartoffel → Kartoffeln
• Süßkartoffel → Süßkartoffeln

So this is a normal plural pattern.

Because this sentence describes a change/result: the vegetables are being cut into small pieces.

With two-way prepositions like in, German uses:

• accusative for direction or change
• dative for location or state

So:

• in kleine Stücke schneiden = cut into small pieces
• in kleinen Stücken = in small pieces, describing a state or form

Here the idea is not where the vegetables are, but what they are being turned into. That is why accusative is used: in kleine Stücke.

Because kleine is an adjective before a plural noun with no article, and the phrase is in the accusative plural.

The noun is:

• Stücke = plural of Stück

Since there is no article before it, the adjective takes a strong ending. In the accusative plural, that ending is -e:

• kleine Stücke
• frische Tomaten
• große Teller

So kleine is the correct form here.

Yes, often you can.

• für das Abendessen emphasizes purpose: this is intended for dinner
• zum Abendessen is also very common and often sounds a bit more natural in everyday speech: for dinner / at dinnertime

So both can work, depending on context.

For example:

• Für das Abendessen schneide ich ...
• Zum Abendessen schneide ich ...

The difference is small in many situations.

Yes. German word order is flexible, but the verb-second rule in main clauses still applies.

You can say:

• Für das Abendessen schneide ich eine Aubergine und zwei Süßkartoffeln in kleine Stücke.
• Ich schneide für das Abendessen eine Aubergine und zwei Süßkartoffeln in kleine Stücke.
• Ich schneide eine Aubergine und zwei Süßkartoffeln für das Abendessen in kleine Stücke.

These all mean roughly the same thing, but the emphasis changes a little.

Putting Für das Abendessen first highlights the dinner context. That is why it appears first in your original sentence.