Jeg vet ikke hvor mye som er igjen i kjøleskapet.

Vet is the present tense of vite, which is used for knowing a fact, piece of information, or answer.

So jeg vet ikke means I don’t know in the sense of I don’t know the answer / I’m not sure.

Norwegian also has kjenne, but that means know in the sense of being familiar with someone or something:

• Jeg kjenner ham. = I know him.
• Jeg kjenner byen godt. = I know the city well.

In your sentence, the speaker is talking about not knowing an amount, so vite is the correct verb.

In a main clause, Norwegian usually puts the finite verb before ikke.

So:

• Jeg vet ikke = I do not know

This is normal Norwegian word order:

• Jeg kommer ikke.
• Han spiser ikke kjøtt.

A useful contrast: in subordinate clauses, ikke usually comes before the finite verb:

• ... fordi jeg ikke vet = ... because I don’t know

So the position of ikke helps show whether you are in a main clause or a subordinate clause.

Hvor mye means how much, and it is used for an amount rather than a number of countable items.

That fits this sentence because the speaker is asking about the amount of something left in the fridge, not necessarily counting individual objects.

Use:

• hvor mye = how much
• hvor mange = how many

Compare:

• Jeg vet ikke hvor mye som er igjen i kjøleskapet.
  = amount / quantity

• Jeg vet ikke hvor mange egg som er igjen i kjøleskapet.
  = number of eggs

So if you are talking about a general quantity of food, drink, etc., hvor mye is the natural choice.

Here, som introduces the embedded clause because hvor mye is functioning as the subject-like element of that clause.

This is similar to other Norwegian patterns:

• Jeg vet hvem som kommer. = I know who is coming.
• Jeg vet hva som skjedde. = I know what happened.
• Jeg vet hvor mye som er igjen. = I know how much is left.

A helpful rule of thumb:

• if the question word is the subject of the embedded clause, Norwegian often uses som
• if it is not the subject, som is usually not used

Compare:

• Jeg vet hva som skjedde.
  (what = subject of skjedde)

• Jeg vet hva du kjøpte.
  (what = object of kjøpte, so no som)

Because after jeg vet ikke, the question becomes an embedded question (also called an indirect question), not a direct question.

Direct question:

• Hvor mye er igjen i kjøleskapet?

Embedded question:

• Jeg vet ikke hvor mye som er igjen i kjøleskapet.

In embedded questions, Norwegian does not use normal question word order. Instead, it uses subordinate-clause order.

So English does something similar:

• Direct: How much is left?
• Embedded: I don’t know how much is left.

Norwegian follows the same basic idea, though with som here as well.

Yes, igjen can often mean again, but in this sentence it means left or remaining.

With være, the expression være igjen means:

• to be left
• to remain

So:

• er igjen = is left / remains

Examples:

• Det er litt melk igjen. = There is a little milk left.
• Er det noe mat igjen? = Is there any food left?

So in this sentence, igjen does not mean again.

Norwegian does not always use a dummy subject in the same way English does.

English often says:

• how much there is left

But Norwegian can simply say:

• hvor mye som er igjen

The amount itself is enough to structure the clause. So Norwegian does not need to mirror English there here.

That said, Norwegian sometimes does use det in other similar expressions, but in this sentence hvor mye som er igjen is perfectly natural and idiomatic.

I means in, so i kjøleskapet means in the refrigerator.

Use i because something is inside the fridge. If you said på kjøleskapet, that would mean on the refrigerator.

As for kjøleskapet:

• et kjøleskap = a refrigerator
• kjøleskapet = the refrigerator

The ending -et is the definite article attached to the noun. That is very normal in Norwegian.

So instead of a separate word like English the, Norwegian often puts definiteness at the end of the noun.

Not if you want to ask about the amount.

• hvor mye = how much
• hva = what

These ask different things:

• Jeg vet ikke hvor mye som er igjen i kjøleskapet.
  = I don’t know how much is left in the fridge.

• Jeg vet ikke hva som er igjen i kjøleskapet.
  = I don’t know what is left in the fridge.

So hva asks about the identity of the thing(s), while hvor mye asks about the quantity.

That is an important difference.