Vi købte en høj reol, fordi bøgerne lå på gulvet i hele rummet.

Because reol is a common-gender noun in Danish, so it takes en, not et.

The adjective høj is also in the correct form for:

• indefinite
• singular
• common-gender

So:

• en høj reol = correct
• et højt skab = with a neuter noun, the adjective would usually get -t
• høje reoler = plural

So the pattern here is very typical Danish adjective agreement.

Reol usually means a shelving unit, bookcase, or set of shelves.

In this sentence, it most naturally means a bookcase / tall shelf unit, since the reason given is that the books were on the floor. Danish can also say bogreol for a more explicitly book-related shelf, but reol by itself is very common.

In Danish, definiteness is often shown by adding an ending to the noun.

Here is the pattern:

• bog = book
• bøger = books
• bøgerne = the books

So Danish often puts the at the end of the noun instead of using a separate word like English does.

Danish often prefers a position verb instead of a general verb like to be.

Here, lå is the past tense of ligge, meaning to lie or to be lying / located in a lying position. Since books on the floor are naturally imagined as lying there, lå is the idiomatic choice.

So:

• Bøgerne lå på gulvet = the books were lying on the floor

Using var would be grammatically possible in some contexts, but it would sound less natural here.

This is a very common Danish learner question.

• ligge = to lie / to be lying
  
  • intransitive: no direct object
• lægge = to lay / to put
  
  • transitive: takes a direct object

Examples:

• Bøgerne lå på gulvet. = The books were lying on the floor.
• Jeg lagde bøgerne på gulvet. = I put the books on the floor.

In your sentence, lå is from ligge, because the books are not doing the action of being placed by someone in that clause; they are simply in that position.

Both are in the simple past (also called the preterite).

• købe → købte = bought
• ligge → lå = lay / were lying

So the whole sentence is describing a past situation and a past action.

A useful contrast:

• Vi købte ... = We bought ...
• Vi har købt ... = We have bought ...

Because fordi introduces a subordinate clause.

In a standard Danish subordinate clause, the word order is usually:

• conjunction + subject + verb + ...

So:

• fordi bøgerne lå på gulvet ...

That is different from main-clause word order in Danish, where the finite verb is normally in second position.

For example:

• Main clause: Bøgerne lå på gulvet.
• Subordinate clause: ... fordi bøgerne lå på gulvet.

På gulvet means on the floor.

The noun is:

• et gulv = a floor
• gulvet = the floor

So gulvet is the definite singular form of gulv.
The preposition på means on.

Together:

• på gulvet = on the floor

Literally, it means in the whole room, but in natural English it often means something like all over the room or throughout the room.

The idea is that the books were not just in one small spot; they were spread around the room.

So i hele rummet gives a stronger sense of the mess being widespread.

With hele meaning whole / entire, Danish normally says:

• hele rummet = the whole room
• hele huset = the whole house
• hele dagen = the whole day

This is a normal pattern. You do not usually add a separate definite article before the noun here.

Also, det hele is a different expression meaning the whole thing / all of it, so it is not the same structure.

Danish høj can correspond to both high and tall in English, depending on context.

For people and upright objects, English often prefers tall:

• en høj mand = a tall man
• en høj reol = a tall bookcase / tall shelving unit

So the Danish word stays the same, but English chooses the most natural translation based on the noun.

It depends on which Danish comma system is being used.

In modern Danish, both of these can be accepted:

• Vi købte en høj reol, fordi bøgerne lå på gulvet i hele rummet.
• Vi købte en høj reol fordi bøgerne lå på gulvet i hele rummet.

Many writers do put a comma before a subordinate clause introduced by fordi, as in your sentence. So what you see here is very normal.

For an English speaker, ø is one of the trickier Danish vowels.

A rough guide:

• ø is a front rounded vowel — say something a bit like eh or uh, but with rounded lips
• høj has øj, which sounds somewhat like the vowel in English boy, but not exactly

Very rough approximations:

• købte ≈ something like KURP-tuh or KYUP-tuh, but neither is exact
• bøgerne ≈ BUR-gher-neh with rounded lips on the first vowel
• høj ≈ somewhat like hoy

The best approach is to listen to native audio and imitate it, because Danish vowels are hard to learn accurately from spelling alone.