Køkkenet ser rodet ud, så jeg tager en kost og begynder at feje.

Because Danish puts the definite article at the end of the noun.

• et køkken = a kitchen
• køkkenet = the kitchen

So -et is the definite ending for a singular neuter noun. The basic noun is et køkken.

Se ud is a very common Danish expression meaning to look / to appear.

So:

• Køkkenet ser rodet ud = The kitchen looks messy

You will see this pattern a lot:

• Hun ser glad ud = She looks happy
• Det ser godt ud = It looks good

Because se ud behaves like a particle verb. In a main clause, the finite verb comes early, but the particle often stays later in the sentence.

So Danish says:

• ser rodet ud
• not ser ud rodet

The description (rodet) goes between ser and ud.

Yes.

• Køkkenet er rodet = The kitchen is messy
• Køkkenet ser rodet ud = The kitchen looks messy

The difference is small:

• er rodet states the condition more directly
• ser rodet ud focuses on how it appears

In everyday use, both can work, but ser ... ud sounds a bit more like an observation.

Rodet is an adjective meaning messy, cluttered, untidy, disorganized.

In this sentence it is a predicate adjective, because it describes the subject after ser ... ud.

It is best to learn rodet as a whole adjective. A learner might expect something more transparent, but rodet is the normal word you use for a messy room, kitchen, desk, and so on.

Here så means so or therefore.

It links the two ideas:

• The kitchen looks messy
• so I take a broom and start sweeping

Danish så can mean several things in other contexts, such as then or so, but here it clearly expresses a result.

Because in this sentence så is being used like a coordinating conjunction joining two main clauses.

So the word order stays:

• ..., så jeg tager en kost ...

But if Så starts a new main clause by itself, then Danish uses normal V2 word order, so the verb comes before the subject:

• Så tager jeg en kost. = Then/So I take a broom.

So both are possible, but they are structured a little differently.

Danish often uses the simple present for actions happening right now, for a near-future action, or for a narrative/live description.

So jeg tager en kost can naturally mean:

• I take a broom
• I grab a broom
• I’m picking up a broom

In English, we often prefer a more specific tense, but Danish simple present is very flexible.

Literally, yes. But in natural English, the best translation in this context is usually:

• grab a broom
• pick up a broom
• get a broom

Danish tage is a very broad verb, and English often uses a more specific verb depending on the situation.

Because kost is a common-gender noun, so the indefinite article is en:

• en kost = a broom

And it is indefinite because the sentence is introducing it as just a broom, not the broom.

• en kost = a broom
• kosten = the broom

Because begynde normally takes at + infinitive.

So:

• begynder at feje = begin/start to sweep
• begynder at læse = begin/start to read

This is different from modal verbs, which do not use at:

• kan feje
• vil feje
• skal feje

Yes.

• ... og fejer = ... and sweep / and am sweeping
• ... og begynder at feje = ... and start sweeping

The version with begynder at emphasizes the start of the action. The shorter version just states the action itself.

Because the sentence contains two main clauses:

• Køkkenet ser rodet ud
• så jeg tager en kost og begynder at feje

In standard Danish punctuation, a comma is commonly used between coordinated main clauses like this. So the comma helps mark the break between the observation and the result.