Beiß nicht in den Apfel.

Because this is a command in German, specifically the informal singular imperative. When you give a command to one person you normally leave out du.

• du beißt = you bite
• Beiß! = Bite!

So Beiß nicht in den Apfel. is a command addressed to one person, and the du is simply understood.

Beiß is the imperative form of beißen for du.

A useful pattern is:

• infinitive: beißen
• du form: du beißt
• command to one person: beiß!

So German takes the du form and removes the ending -t. That gives beiß.

Compare:

• du kommst → komm!
• du liest → lies!
• du beißt → beiß!

The letter ß is called Eszett or scharfes S. It is pronounced like ss.

In beißen, the ei is a diphthong, and after a long vowel or diphthong German normally writes ß, not ss. That is why you get:

• beißen
• du beißt
• beiß!

A small extra note: in Switzerland, people usually write ss instead of ß, so you may also see Beiss nicht in den Apfel.

Because all nouns are capitalized in German.

So:

• der Apfel
• ein Apfel
• den Apfel

This is a general rule, not something special about this sentence.

German uses nicht to negate a verb, an action, or a whole clause. It uses kein mainly to negate a noun phrase with an indefinite idea, like a/an or any.

Here, the speaker is telling someone not to do the action of biting into the apple, so nicht is the right word:

• Beiß nicht in den Apfel.

Compare:

• Iss keinen Apfel. = Don't eat an apple / don't eat any apple.
• Beiß nicht in den Apfel. = Don't bite into the apple.

So nicht is used because the negation is aimed at the action.

In commands, nicht often comes after the imperative verb.

So the normal pattern is:

• Verb + nicht + ...

That gives:

• Beiß nicht ...
• Geh nicht ...
• Mach das nicht ...

Its exact position can depend on what is being negated, but in a sentence like this, Beiß nicht in den Apfel. is the natural word order.

Because German usually expresses this idea with the preposition in: in etwas beißen = to bite into something.

English often says either:

• bite the apple
• bite into the apple

German strongly prefers the version with in here:

• in den Apfel beißen

That is the normal idiomatic way to say that you sink your teeth into the apple.

Because in is a two-way preposition in German. That means it can take either:

• accusative for direction/movement
• dative for location

Here the idea is into the apple, so German uses the accusative:

• in den Apfel

Since Apfel is masculine:

• nominative: der Apfel
• accusative: den Apfel
• dative: dem Apfel

So den is correct because the phrase expresses movement or direction into something.

You learn that from its dictionary form:

• der Apfel

Once you know that, the article changes depending on case:

• nominative: der Apfel
• accusative: den Apfel
• dative: dem Apfel

So in this sentence, den Apfel tells you that Apfel is masculine and is in the accusative case.

Yes, an exclamation mark is very common with commands:

• Beiß nicht in den Apfel!

But German does not absolutely require it every time. Without the exclamation mark, the sentence can look a little less forceful or simply more neutral in writing. Both are possible.