Hinter der Hecke liegt ein Ast, und darunter kommt eine dicke Wurzel aus dem Boden.

Because hinter is a two-way preposition. That means it can take:

• dative for a location: behind the hedge
• accusative for movement toward that position: to behind the hedge

In this sentence, the branch is already located behind the hedge, so German uses the dative:

• hinter der Hecke = behind the hedge

If something were being moved there, you would use accusative:

• Ich lege den Ast hinter die Hecke. = I put the branch behind the hedge.

German main clauses follow the verb-second rule. The finite verb must come in the second position.

Here, the sentence starts with the adverbial phrase:

• Hinter der Hecke

That takes the first position, so the verb must come next:

• Hinter der Hecke liegt ein Ast.

The subject ein Ast comes after the verb.

You could also say:

• Ein Ast liegt hinter der Hecke.

That is also correct, but it puts the focus differently. Starting with Hinter der Hecke emphasizes the location.

Ein Ast is in the nominative case because it is the subject of the verb liegt.

So in:

• Hinter der Hecke liegt ein Ast

the thing that is lying there is ein Ast, so it is the subject.

Darunter means under it, underneath it, or sometimes under there, depending on context.

Here it refers back to the previously mentioned location/object, especially the hedge area:

• darunter = under that / beneath it

German often uses these da- compounds instead of repeating a noun or pronoun after a preposition.

So instead of something like repeating unter der Hecke, German can say:

• darunter

This sounds natural and avoids repetition.

For the same reason as in the first clause: verb-second word order.

The second clause begins with darunter, so that takes the first position. The verb must then come second:

• darunter kommt eine dicke Wurzel aus dem Boden

The subject eine dicke Wurzel comes after the verb.

You could also say:

• Eine dicke Wurzel kommt darunter aus dem Boden.

That is grammatically possible, but the original version sounds more natural if you want to keep the focus on the place.

In German, kommen is often used more broadly than English come. It can describe something appearing, emerging, or sticking out from somewhere.

So:

• eine Wurzel kommt aus dem Boden
  means something like
• a root comes out of the ground
• a root emerges from the ground
• a root sticks out of the ground

This is idiomatic German. English would often prefer comes out, emerges, or protrudes.

Because aus always takes the dative case.

The noun is:

• der Boden = the ground / soil / floor

Its dative singular form is:

• dem Boden

So:

• aus dem Boden = out of the ground

It is nominative singular, because it is the subject of kommt.

In the clause:

• darunter kommt eine dicke Wurzel aus dem Boden

the thing that is coming out of the ground is eine dicke Wurzel, so that noun phrase is nominative.

The adjective ending -e is used because:

• Wurzel is feminine
• it is singular
• it is in the nominative
• it follows the indefinite article eine

So the pattern is:

• eine dicke Wurzel

This is a standard adjective ending in German.

Compare:

• ein dicker Ast — masculine nominative
• eine dicke Wurzel — feminine nominative
• ein dickes Buch — neuter nominative

This is a very common question.

• liegen = to lie / to be lying
  describes a position; it does not take a direct object
• legen = to lay / to put
  describes an action; it does take a direct object

In your sentence:

• liegt ein Ast = a branch is lying

That is a state or position.

Compare:

• Der Ast liegt hinter der Hecke. = The branch is lying behind the hedge.
• Ich lege den Ast hinter die Hecke. = I put the branch behind the hedge.

In context, it most naturally suggests beneath that area / beneath it, connected to the hedge and the scene just described.

German darunter can be a little flexible in English translation. Depending on context, it may mean:

• under it
• beneath it
• underneath
• below that

Here, it helps the sentence move from one visible thing to another in the same scene: first the branch behind the hedge, then below that, a thick root coming out of the ground.

Yes. Und simply links the two clauses, and darunter gives a clear spatial connection: first one thing is mentioned, then something below that.

So the structure is:

• Hinter der Hecke liegt ein Ast,
• und darunter kommt eine dicke Wurzel aus dem Boden.

This is a natural descriptive sentence, especially in written or visual description.

Yes. Wurzel can mean:

• a root of a plant
• the root of something in a figurative sense
• in math, a square root

But in this sentence, because of aus dem Boden and the outdoor setting, it clearly means a plant/tree root.