Jeg drikker kaffe kun hvis jeg er meget træt.

Because Danish, like English, normally needs a subject in each clause.

This sentence has:

• a main clause: Jeg drikker kaffe
• a subordinate clause: hvis jeg er meget træt

So jeg appears once in each clause:

• Jeg drikker kaffe = I drink coffee
• hvis jeg er meget træt = if I am very tired

You cannot usually leave out the second jeg.

Drikker is the present tense form of drikke.

• drikke = to drink
• drikker = drink / am drinking

In Danish, the present tense often ends in -r:

• jeg drikker = I drink
• jeg spiser = I eat
• jeg læser = I read

Drik is the imperative, meaning drink!

Because with substances and general nouns, Danish often does not use an article where English might.

So:

• Jeg drikker kaffe = I drink coffee

This means coffee in a general sense, not one specific coffee.

Compare:

• Jeg drikker kaffe = I drink coffee
• Jeg drikker en kaffe = I am drinking a coffee / one coffee

In your sentence, the general idea is meant, so kaffe appears without en.

Here kun means only, and it limits the whole condition.

So kun hvis jeg er meget træt means only if I am very tired.

That placement is natural because the speaker is saying:

• I drink coffee, but only under this condition.

So the structure is:

• Jeg drikker kaffe
  • kun hvis ...

If you move kun, the emphasis can change. For example:

• Jeg drikker kun kaffe, hvis jeg er meget træt can sound more like I only drink coffee if I am very tired, which may suggest coffee is the only thing I drink in that situation, or simply be interpreted similarly depending on context.

Your original sentence clearly makes only if the key idea.

Because in Danish subordinate clauses, the subject normally comes before the verb, just as in English.

So:

• hvis jeg er meget træt = if I am very tired

The order is:

• hvis = if
• jeg = I
• er = am
• meget træt = very tired

Danish does have verb-second word order in many main clauses, but that does not apply in the same way after a subordinating conjunction like hvis.

Hvis means if and introduces a condition.

So the sentence says that drinking coffee happens only under a certain condition:

• hvis jeg er meget træt = if I am very tired

It is a subordinating conjunction, which means it introduces a subordinate clause rather than a full independent statement.

Because træt is an adjective, meaning tired, and Danish often uses to be + adjective just like English.

So:

• jeg er træt = I am tired

This is not the action to tire someone; it is a description of the person's state.

Also, træt does not change here because it describes jeg, and predicate adjectives after er usually stay in their basic form in this kind of sentence.

Meget means very, and like in English, it comes before the adjective it modifies.

So:

• meget træt = very tired

This is a common pattern:

• meget glad = very happy
• meget stor = very big
• meget vigtigt = very important

Yes. You can also say:

Kun hvis jeg er meget træt, drikker jeg kaffe.

This means the same basic thing, but the emphasis is slightly different. Starting with the condition makes it more prominent.

Notice the word order in the main clause after the fronted subordinate clause:

• drikker jeg kaffe not
• jeg drikker kaffe

That happens because Danish main clauses usually have the finite verb in second position.

By default, jeg drikker in the present tense often expresses a general habit or rule:

• Jeg drikker kaffe kun hvis jeg er meget træt = I drink coffee only if I am very tired

So it sounds like a general statement about the speaker’s behavior.

If you wanted to make it more clearly about the present moment, context would usually do that. Danish often uses the simple present where English might also use am drinking in some contexts.

Yes, it is a very common combination meaning only if.

Examples:

• Jeg går kun hvis du går med. = I’m only going if you come too.
• Han køber den kun hvis den er billig. = He buys it only if it is cheap.

So in your sentence, kun hvis should be understood together as a unit expressing a limited condition.