Brillestellet ligger i brilleetuiet.

Because Danish often makes nouns definite by adding the equivalent of the to the end of the noun.

• et brillestel = a glasses frame
• brillestellet = the glasses frame

And:

• et brilleetui = a glasses case
• brilleetuiet = the glasses case

The ending -et is used here because these are neuter nouns that normally take et in the indefinite form.

So instead of saying a separate word for the, Danish usually attaches it to the noun.

In Danish, the definite article is very often a suffix, not a separate word.

So English:

• the frame
• the case

becomes Danish:

• stellet
• etuiet

This is one of the big differences from English. Danish does have separate definite words like den, det, and de, but in a simple sentence like this, the normal pattern is the noun with a definite ending attached.

Brillestellet is a compound noun:

• brille = glasses / eyeglasses
• stel = frame

So brillestel means glasses frame or spectacle frame.

Then brillestellet means the glasses frame.

A learner might expect briller because glasses in English is plural, but Danish often uses the singular form inside compounds. That is normal.

Because Danish compound nouns usually use the basic dictionary form of the first noun, often singular.

So even though you normally say:

• briller = glasses

you still get compounds like:

• brillestel = glasses frame
• brilleetui = glasses case
• brillebutik = optician's shop / glasses shop

This is very common in Danish. English sometimes does something similar too, as in toothbrush rather than teethbrush.

Danish often uses a position verb where English would often just use is.

Here, ligger comes from at ligge, which means to lie. It is used for something that is located somewhere in a lying or resting position.

So:

• Brillestellet ligger i brilleetuiet. literally feels like
• The frame lies in the glasses case.

In natural English, you would usually just say The frame is in the glasses case, but Danish prefers ligger here because it describes how the object is situated.

This is a very common question, because Danish uses these location verbs much more than English.

A simple guide:

• ligge = to lie, for things that are flat, horizontal, or just resting somewhere
• stå = to stand, for things that are upright
• sidde = to sit, for things attached, seated, or positioned in a certain way

Examples:

• Bogen ligger på bordet.
  The book is on the table.

• Flasken står på bordet.
  The bottle is on the table.

• Billedet sidder på væggen.
  The picture is on the wall.

In your sentence, a glasses frame inside a case is naturally thought of as lying, so ligger is the normal choice.

Yes, people would understand it, but ligger is more natural and idiomatic.

• er i simply says it is in the case
• ligger i says it is located there in a lying/resting position

Danish often prefers the more specific position verb, especially for physical objects. So er is not exactly wrong, but ligger sounds better in everyday Danish.

I means in.

So:

• i brilleetuiet = in the glasses case

That is the normal preposition because the frame is inside the case, not just on it or near it.

Compare:

• i = in
• på = on
• ved = by / near

So i is exactly what you would expect for something inside a container.

Yes. Danish very often writes compound nouns as one word, where English often uses two words.

So:

• brilleetui = glasses case
• brillestel = glasses frame

This is a very important habit to get used to in Danish. If English would naturally say X case, X shop, X frame, X door, and so on, Danish often makes it a single compound word.

Ligger is the present tense form of at ligge.

So:

• at ligge = to lie
• ligger = lies / is lying

Danish verbs are nice in one way: they do not change according to the subject the way English does.

So you get:

• jeg ligger = I am lying
• du ligger = you are lying
• den ligger = it is lying
• de ligger = they are lying

The form ligger stays the same.

It is a long compound, so it helps to break it up:

• brille
  • etui
    • -et

So think of it in parts rather than as one giant block.

A rough English-friendly guide would be something like:

• brille ≈ BRIL-le
• etui ≈ eh-TU-i
• brilleetuiet ≈ BRIL-le-eh-TU-i-eth

That said, Danish pronunciation is often less clear-cut than spelling suggests, so the best approach is to learn the word in chunks and listen to native audio if possible.

In this sentence, both key nouns are neuter nouns:

• et brillestel
• et brilleetui

That is why their definite form ends in -et:

• brillestellet
• brilleetuiet

Unfortunately, you often cannot reliably guess noun gender just from the meaning. Usually, you need to learn the noun together with its article:

• et brillestel
• et brilleetui

That is the safest habit in Danish.