Når kagen er færdig, tager vi den ud af ovnen.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

In this sentence, the whole clause Når kagen er færdig comes first, so it takes the first slot. That means the verb in the main clause must come next:

• Når kagen er færdig, tager vi den ud af ovnen.

If you put the time clause at the end instead, the word order becomes more familiar:

• Vi tager den ud af ovnen, når kagen er færdig.

So both are correct, but when the når-clause comes first, you get tager vi.

Når means when here, but specifically in the sense of:

• something that happens in the future
• or something that happens whenever / every time

So Når kagen er færdig means when the cake is ready.

A useful contrast is da, which is usually used about a specific event in the past:

• Da kagen var færdig, tog vi den ud af ovnen.
  = When the cake was ready, we took it out of the oven.

So a simple rule is:

• når = when (future, repeated, general)
• da = when (specific past event)

Danish often uses the present tense for things that are in the future, especially when the time is already clear from the context.

So:

• Når kagen er færdig, tager vi den ud af ovnen.

is completely natural, even though the cake may not be ready yet.

English sometimes does something similar:

• When the cake is ready, we take it out.
• more idiomatically: When the cake is ready, we’ll take it out.

Danish does not need a separate future form here.

Kagen means the cake. Danish usually makes the definite form by adding the article to the end of the noun.

So:

• en kage = a cake
• kagen = the cake

This is one of the most important differences from English. Instead of a separate word like the, Danish often attaches the definite ending directly to the noun.

Because kage is a common-gender noun, and common-gender nouns take den when replaced by a pronoun.

• en kage → den
• et brød → det

So:

• Når kagen er færdig, tager vi den ud af ovnen.

If the noun were neuter, you would use det instead.

Because den is the direct object of tager and refers back to kagen.

Danish, like English, often avoids repeating the noun:

• Når kagen er færdig, tager vi kagen ud af ovnen.
  This is grammatical, but repetitive.

More natural:

• Når kagen er færdig, tager vi den ud af ovnen.

So den works just like it in English.

Ud af means out of.

It is a very common combination:

• ud = out
• af = of / from
• ud af = out of

So:

• tage den ud af ovnen = take it out of the oven

This expression is used for movement from inside something to outside it:

• ud af huset = out of the house
• ud af bilen = out of the car

Ovnen is the definite form of ovn, meaning the oven.

• en ovn = an oven
• ovnen = the oven

Again, Danish usually marks definiteness with an ending on the noun rather than a separate word like the.

So:

• ud af ovnen = out of the oven

Because færdig agrees with the noun it describes.

Here, kagen is:

• singular
• common gender

So the adjective stays in its basic form:

• kagen er færdig

Compare:

• brødet er færdigt
  because brødet is neuter
• kagerne er færdige
  because kagerne is plural

So the pattern is:

• common singular: færdig
• neuter singular: færdigt
• plural: færdige

Because Når kagen er færdig is a subordinate clause placed before the main clause.

The comma marks the boundary between:

• the introductory when-clause
• the main statement

So the structure is:

• Når kagen er færdig, = subordinate clause
• tager vi den ud af ovnen. = main clause

This is very common in written Danish.

Yes. That is also completely correct.

The difference is mainly one of focus and word order:

• Når kagen er færdig, tager vi den ud af ovnen.
  starts with the time condition
• Vi tager den ud af ovnen, når kagen er færdig.
  starts with the main action

Both mean the same thing. The first version is often used when you want to emphasize when the action happens.