Synsvinklen ændrer, hvordan vi fortolker budskabet.

Because synsvinkel is a singular noun in the definite form.

• en synsvinkel = a point of view / a perspective
• synsvinklen = the point of view / the perspective

In Danish, the definite article is usually added to the end of the noun instead of being a separate word before it.

Because budskab is a neuter noun.

• et budskab = a message
• budskabet = the message

So the sentence shows two different noun genders:

• en synsvinkel → synsvinklen
• et budskab → budskabet

This is very normal in Danish.

Ændrer is the present tense of at ændre.

• at ændre = to change
• ændrer = change / changes

Danish present tense is simple: the verb usually gets -r, and it does not change for different persons.

• jeg ændrer
• du ændrer
• vi ændrer

So synsvinklen ændrer means that the point of view changes or affects/changes something.

Because this clause needs a finite verb, not an infinitive.

• at fortolke = to interpret
• fortolker = interpret / interprets

In hvordan vi fortolker budskabet, the subject is vi, so the verb must be a normal tense form. Here it is present tense: fortolker.

If you used fortolke, it would sound unfinished, because fortolke is just the infinitive.

Because at is used with an infinitive, and fortolker is not an infinitive.

Compare:

• at fortolke budskabet = to interpret the message
• vi fortolker budskabet = we interpret the message

After hvordan, you have a full subordinate clause with its own subject and verb, so at is not used.

Hvordan introduces the subordinate clause hvordan vi fortolker budskabet.

It means how, and the whole clause functions like the thing being affected by ændrer:

• Synsvinklen ændrer what?
• hvordan vi fortolker budskabet

So the clause acts like the object or content of the verb.

Because this is an embedded question / subordinate clause, not a main question.

In a main question, Danish often has verb-before-subject order:

• Hvordan fortolker vi budskabet? = How do we interpret the message?

But inside a subordinate clause, Danish usually keeps normal subject-verb order:

• hvordan vi fortolker budskabet

So:

• main question: Hvordan fortolker vi ... ?
• embedded clause: hvordan vi fortolker ...

The comma marks the beginning of the subordinate clause hvordan vi fortolker budskabet.

In Danish, many writers put a comma before subordinate clauses. This is especially common in the traditional start comma system. So the comma helps show that the sentence has:

• a main clause: Synsvinklen ændrer
• a subordinate clause: hvordan vi fortolker budskabet

You may also see Danish written without this comma, depending on the comma system being used.

Not here.

Ændre sig means to change in the sense of changing itself:

• Situationen ændrer sig = The situation is changing

But ændre without sig can take an object:

• Det ændrer planen = That changes the plan

In your sentence, ændrer affects the whole clause hvordan vi fortolker budskabet, so sig is not needed.

Yes.

• ændrer = present tense
• fortolker = present tense

This is very natural in Danish when stating a general truth or a general fact. The sentence is not necessarily about something happening right now at this exact moment; it can also express a general principle.

It is singular definite: synsvinklen = the point of view.

Useful forms are:

• en synsvinkel = a point of view
• synsvinklen = the point of view
• synsvinkler = points of view
• synsvinklerne = the points of view

So in this sentence, it refers to one specific viewpoint or perspective.