Mein Bruder möchte keinen Blumenkohl, sondern lieber Brokkoli mit etwas Sahne.

Because mein is a possessive determiner, and it changes its ending depending on case, gender, and number.

Here, Bruder is:

• masculine
• singular
• nominative (the subject of the sentence)

In that combination, the correct form is mein:

• mein Bruder = my brother

Compare:

• mein Bruder = nominative
• meinen Bruder = accusative
• meinem Bruder = dative

So meiner Bruder would be incorrect here.

Möchte means would like or would want.

It is much softer and more polite than will.

• Ich will Brokkoli. = I want broccoli.
  This can sound quite direct or forceful.
• Ich möchte Brokkoli. = I would like broccoli.
  This sounds more natural and polite in many situations.

So in this sentence, möchte expresses preference in a gentle way.

Because in a normal German main clause, the finite verb goes in second position.

Here the sentence begins with the subject:

• Mein Bruder = position 1
• möchte = position 2

Then the rest follows:

• keinen Blumenkohl, sondern lieber Brokkoli mit etwas Sahne

This is one of the most important German word-order rules: verb second in main clauses.

Because Blumenkohl is masculine:
der Blumenkohl

And in this sentence it is the direct object of möchte, so it must be in the accusative case.

The negative article kein changes like ein, and the masculine accusative form is:

• keinen

So:

• der Blumenkohl → nominative
• keinen Blumenkohl → accusative after möchte

Use kein to negate a noun that would otherwise have ein or no article.

So:

• Blumenkohl = cauliflower
• kein Blumenkohl = no cauliflower / not any cauliflower

Use nicht to negate other things, such as:

• a verb
• an adjective
• an adverb
• or the whole sentence

So here, since the speaker is rejecting the noun Blumenkohl, kein is the correct choice:

• Mein Bruder möchte keinen Blumenkohl.

Not:

• Mein Bruder möchte nicht Blumenkohl.
  This is not the normal way to say it.

In German, food names are often used without an article when you mean them in a general or uncountable sense.

So:

• Ich esse gern Brokkoli.
• Sie trinkt Kaffee.
• Er möchte Reis.

Here, Brokkoli means broccoli as food, not one whole broccoli head.

That is why lieber Brokkoli sounds natural.

If you wanted to talk about one whole item, you might use an article in some contexts, but that is not the idea here.

Sondern means but rather and is used after a negative idea to replace it with the correct alternative.

Pattern:

• not X, but rather Y

That is exactly what happens here:

• keinen Blumenkohl, sondern lieber Brokkoli
• not cauliflower, but rather broccoli

This is different from aber, which usually just means but and does not necessarily correct a negative statement.

So after keinen Blumenkohl, sondern is the right conjunction.

Here lieber means rather or preferably.

It shows preference:

• Blumenkohl? No. Brokkoli instead.

So:

• sondern lieber Brokkoli
  = but rather broccoli = but preferably broccoli

Lieber is related to preference, like prefer in English.

Examples:

• Ich trinke lieber Tee als Kaffee. = I prefer tea to coffee.
• Er möchte lieber zu Hause bleiben. = He would rather stay at home.

Because German normally uses a comma before sondern.

This is standard punctuation:

• ..., sondern ...

Even when sondern connects smaller parts of a sentence rather than two full clauses, the comma is still usually there.

So:

• Mein Bruder möchte keinen Blumenkohl, sondern lieber Brokkoli ...

is correctly punctuated.

It means with some cream or with a little cream.

• mit = with
• etwas = some / a little
• Sahne = cream

Etwas is very common for talking about a small, indefinite amount:

• etwas Milch = some milk
• etwas Zucker = some sugar
• etwas Wasser = some water

So mit etwas Sahne is a natural way to describe a small amount of cream.

Mit always takes the dative case.

So the whole phrase mit etwas Sahne is in the dative.

You do not see an obvious special ending on Sahne here because:

• Sahne is a feminine singular noun
• there is no article like der or einer showing the case
• the noun itself stays the same in form here

Compare:

• mit der Sahne = with the cream
• mit etwas Sahne = with some cream

So the case is still dative, even though the noun form does not visibly change.

Because in German, all nouns are capitalized.

So in this sentence:

• Bruder
• Blumenkohl
• Brokkoli
• Sahne

are all nouns, so they begin with a capital letter.

This is a basic spelling rule in German and applies everywhere, not just at the start of a sentence.