Hinter dem Busch liegt ein Ast, und daneben kommt eine Wurzel aus dem Boden.

Because hinter is a two-way preposition. With these prepositions, German uses:

• dative for a location: where something is
• accusative for a direction/change of place: where something is going

Here, the branch is located behind the bush, so German uses the dative:

• hinter dem Busch

Compare:

• Der Ast liegt hinter dem Busch. = the branch is behind the bush
• Ich lege den Ast hinter den Busch. = I put the branch behind the bush

So dem Busch is dative because this is a static location.

Because German main clauses follow the verb-second rule.

That means the finite verb must be in the second position, but the first position can be taken by many things, not just the subject. Here, the sentence starts with the location:

• Hinter dem Busch = position 1
• liegt = position 2
• ein Ast = the subject comes after the verb

So:

• Hinter dem Busch liegt ein Ast.

is completely normal German word order.

You could also say:

• Ein Ast liegt hinter dem Busch.

But starting with the location sounds natural if the speaker wants to describe the scene piece by piece.

They are both nominative, because they are the subjects of their clauses.

In German, the subject does not have to come before the verb. So even though you see:

• liegt ein Ast
• kommt eine Wurzel

the nouns are still the subjects.

You can tell from the articles:

• ein Ast = nominative masculine
• eine Wurzel = nominative feminine

This is an important difference from English: word order is more flexible, so you often need to identify the subject by case, not just by position.

German often prefers a more specific verb than English.

• sein = to be
• liegen = to lie, to be lying, to be located lying down

For objects like a branch that are resting somewhere, liegen is very natural. It gives a clearer picture than ist.

So:

• Hinter dem Busch liegt ein Ast.

sounds more vivid than simply saying the branch is there.

English sometimes does something similar with lying there.

Daneben means next to it, beside it, or right next to that.

It refers back to something already mentioned. In this sentence, it points back to the area near the bush. So after mentioning the bush, the speaker says that beside it there is also a root coming out of the ground.

Grammatically, daneben is a pronominal adverb:

• da-
  • neben

German uses these forms a lot instead of repeating a noun phrase.

Yes, you could. For example:

• Hinter dem Busch liegt ein Ast, und neben dem Busch kommt eine Wurzel aus dem Boden.

That is grammatical. But daneben sounds smoother because it avoids repeating dem Busch.

The difference is roughly:

• neben dem Busch = explicitly next to the bush
• daneben = next to it / beside that

So daneben is a natural choice when the thing being referred to has just been mentioned.

It is a natural German way to describe something emerging or sticking out.

Literally, it is a root comes out of the ground, but idiomatically the idea is:

• a root is coming out of the ground
• a root sticks out of the ground
• a root emerges from the ground

German often uses motion verbs like kommen in places where English might prefer a more descriptive phrase.

So this is not strange German; it is just a place where German and English choose different natural wording.

No. In this sentence, aus is a preposition, not a separable prefix.

So the structure is:

• kommt = the verb
• aus dem Boden = a prepositional phrase meaning out of the ground

Also, aus always takes the dative, which is why you get:

• dem Boden

So this is not the same as a separable verb like aufstehen or mitkommen. It is simply kommen plus the phrase aus dem Boden.

Because aus always takes the dative.

So:

• aus dem Boden
• aus dem Haus
• aus der Erde

Unlike hinter, which can take two different cases depending on meaning, aus is straightforward: it always uses dative.

Because the sentence contains two full main clauses:

• Hinter dem Busch liegt ein Ast
• daneben kommt eine Wurzel aus dem Boden

In German, a comma before und in this kind of structure is often used to make the sentence easier to read. In many cases, with und joining two main clauses, the comma can be optional, especially when it helps clarity.

So the comma here is normal and helps separate the two scene descriptions.

They are:

• der Busch = bush
• der Ast = branch
• die Wurzel = root
• der Boden = ground, soil, floor

Knowing the gender helps you understand the forms in the sentence:

• dem Busch = dative of der Busch
• ein Ast = nominative singular
• eine Wurzel = nominative singular
• dem Boden = dative of der Boden

For a learner, this is useful because article changes often show the noun’s role in the sentence.