Jeg vil klippe billedet ud, så jeg kan hænge det på væggen.

Because vil is a modal verb, and modal verbs in Danish are followed by the infinitive without at.

So:

• jeg vil klippe = I want to cut out
• jeg kan hænge = I can hang

This is similar to English, where you say I can go, not I can to go.

Common Danish modal verbs include:

• kan = can
• vil = want to, will
• skal = must, have to
• må = may, must

All of them normally take a bare infinitive.

Klippe ud means cut out, as in cutting something out of paper, a magazine, a photo, and so on.

It is a particle verb, where the particle ud changes the meaning of the basic verb:

• klippe = cut
• klippe ud = cut out

In Danish, these particles are often written separately in normal sentence structure:

• Jeg klipper billedet ud
• Jeg vil klippe billedet ud

So ud is not random extra information. It is part of the verb meaning.

Because billedet is the definite form, meaning the picture.

The noun is:

• et billede = a picture
• billedet = the picture

Danish usually adds the definite article to the end of the noun instead of using a separate word like English the.

Since the sentence refers to a specific picture, Danish uses billedet.

This is how many neuter nouns form the definite singular in Danish.

Pattern:

• et billede = a picture
• billedet = the picture

The ending -et marks definite singular for a neuter noun.

A few more examples:

• et hus → huset
• et brev → brevet
• et vindue → vinduet

So billedet is just the definite form of et billede.

Here så means so that or in order that.

It introduces the reason or purpose for the first action:

• Jeg vil klippe billedet ud = I want to cut out the picture
• så jeg kan hænge det på væggen = so that I can hang it on the wall

So the second clause explains the purpose of the first one.

Be aware that så can also mean so, then, or such, depending on context. But in this sentence it is best understood as so that.

Because så introduces a subordinate clause here.

In Danish main clauses, the finite verb usually comes in second position:

• Jeg kan hænge det på væggen

But in subordinate clauses, the subject usually comes before the finite verb:

• så jeg kan hænge det på væggen

So this is a normal word-order difference between main clauses and subordinate clauses.

Compare:

• Main clause: Jeg kan hænge det på væggen
• Subordinate clause: ... så jeg kan hænge det på væggen

For the same reason as with vil klippe: kan is also a modal verb, and modal verbs take the infinitive without at.

So:

• jeg kan hænge = I can hang
• not jeg kan at hænge

This is a very important pattern in Danish.

Det refers back to billedet.

Since billedet is a singular neuter noun, the correct pronoun is det:

• et billede
• billedet
• det

So:

• Jeg vil klippe billedet ud = I want to cut out the picture
• så jeg kan hænge det på væggen = so that I can hang it on the wall

This works much like English it.

Because væggen means the wall, and Danish often uses the definite form when talking about a specific, understood object.

• en væg = a wall
• væggen = the wall

So på væggen means on the wall.

In this sentence, the speaker is thinking of a particular wall or just the relevant wall in the situation, so the definite form is natural.

Yes, hænge can be used in more than one way.

Here it is transitive, meaning to hang something:

• Jeg kan hænge det på væggen = I can hang it on the wall

But hænge can also be used intransitively, meaning to be hanging:

• Billedet hænger på væggen = The picture is hanging on the wall

So the same verb can describe either:

• putting something in a hanging position
• or being in that position

Context tells you which meaning is intended.

Because the sentence contains two clauses, and Danish normally uses commas to separate clauses.

Here the comma marks the boundary between:

• Jeg vil klippe billedet ud
• så jeg kan hænge det på væggen

Danish comma rules are generally stricter than English ones in this kind of sentence, so learners often notice more commas than they expect.

The sentence has two parts:

1. Main clause
   Jeg vil klippe billedet ud

2. Subordinate clause of purpose
   så jeg kan hænge det på væggen

A rough breakdown is:

• Jeg = subject
• vil = finite verb
• klippe = infinitive
• billedet = object
• ud = particle

then

• så = conjunction
• jeg = subject
• kan = finite verb
• hænge = infinitive
• det = object
• på væggen = prepositional phrase

So the sentence is built very regularly once you see the two-clause structure.

Yes, that is possible, but it changes the structure slightly.

• så jeg kan hænge det på væggen = so that I can hang it on the wall
• for at hænge det på væggen = in order to hang it on the wall

The version with for at is often used when the subject is the same in both parts of the sentence, as it is here.

So this would also be natural:

• Jeg vil klippe billedet ud for at hænge det på væggen

The original sentence is still completely correct. It just uses a full clause with så jeg kan ... instead of an infinitive construction.