Ich schneide den Ast ab.

Because abschneiden is a separable verb.

The full verb is abschneiden. In a normal main clause, German splits this kind of verb into two parts:

• the main conjugated part goes in the usual verb position: schneide
• the prefix goes to the end: ab

So:

• Ich schneide den Ast ab.

But in the infinitive, it stays together:

• abschneiden

This is very common in German:

• anrufen → Ich rufe dich an.
• aufstehen → Ich stehe auf.

The dictionary form is abschneiden.

In this sentence:

• ich = subject
• schneide = the conjugated form
• ab = the separable prefix

So if you look this verb up in a dictionary, you would look for abschneiden, not just schneiden.

Because Ast is the direct object here, so it takes the accusative case.

The noun Ast is masculine, and the masculine definite article changes like this:

• der = nominative
• den = accusative

In this sentence:

• Ich is doing the action
• den Ast is what is being cut off

So German uses the accusative:

• Ich schneide den Ast ab.

If Ast were the subject, it would be der Ast:

• Der Ast fällt ab.

You usually have to learn the noun together with its article:

• der Ast

German noun gender is not always predictable, so it is best to memorize new nouns as a full package:

• der Ast
• plural: die Äste

That makes it much easier to use the correct article later, like den Ast in the accusative.

Schneide is the 1st person singular present tense form of schneiden.

The present tense endings for schneiden are:

• ich schneide
• du schneidest
• er/sie/es schneidet
• wir schneiden
• ihr schneidet
• sie/Sie schneiden

So schneide matches ich.

It can mean either, depending on context.

German often uses the simple present where English might use either:

• I cut the branch off
• I am cutting the branch off

So Ich schneide den Ast ab can express a present action without needing a separate -ing form.

If the context makes it clear that it is happening right now, English often prefers I am cutting the branch off, but German still just uses the present tense.

Because that word order is wrong in a normal main clause.

With a separable verb in a main clause:

• the conjugated verb part comes in the verb-second position
• the prefix goes to the end

So the correct form is:

• Ich schneide den Ast ab.

Not:

• Ich abschneide den Ast.

However, in some other structures the verb stays together, for example with an infinitive:

• Ich will den Ast abschneiden.

Or in a subordinate clause:

• ..., weil ich den Ast abschneide.

Yes, but the conjugated verb still has to stay in second position in a main clause, and ab still goes to the end.

For example:

• Ich schneide den Ast ab.
• Den Ast schneide ich ab.

Both are correct.

The second version puts more emphasis on den Ast. German often moves elements around for emphasis, but the verb rule remains very important.

Because all nouns are capitalized in German.

So:

• ich = not capitalized in the middle of a sentence
• Ast = capitalized because it is a noun

This is a standard rule in German spelling:

• der Ast
• das Haus
• die Tür

Schneiden means to cut in a general sense.

Abschneiden means to cut off or cut away something from something else.

So:

• Ich schneide den Ast. = I am cutting the branch.
• Ich schneide den Ast ab. = I am cutting the branch off.

The prefix ab- adds the idea of separation.

Not if you want the meaning cut off.

If you remove ab, you change the verb:

• schneiden = to cut
• abschneiden = to cut off

So:

• Ich schneide den Ast. does not mean exactly the same thing as
• Ich schneide den Ast ab.

The second sentence clearly expresses that the branch is being separated from something.

The plural is Äste.

So:

• singular: der Ast
• plural: die Äste

Examples:

• Ich schneide den Ast ab.
• Ich schneide die Äste ab.

Notice that in the plural accusative, the article is die, not den.