I baggrunden kan man se hendes bøger og den blå lampe ved vinduet.

Because Danish main clauses normally follow the verb-second rule.

That means:

• the first position can be filled by many things: the subject, a time phrase, a place phrase, etc.
• the finite verb must then come in second position

Here, I baggrunden is placed first for emphasis, so the verb kan has to come next:

• I baggrunden = first position
• kan = second position
• man = subject
• se = infinitive

So I baggrunden kan man se ... is the normal Danish structure.

If the subject came first, you would get:

• Man kan se hendes bøger og den blå lampe ved vinduet i baggrunden.

That is also grammatical, but it gives a slightly different focus.

Man is an impersonal pronoun in Danish. It often means something like:

• one
• you in a general sense
• people
• sometimes English would use a passive, such as can be seen

So kan man se means something like:

• one can see
• you can see
• can be seen

It does not mean man in the English sense of adult male.

This word is very common in descriptions, instructions, and general statements.

In Danish, i baggrunden is the normal expression for in the background.

A few useful points:

• i is the standard preposition here
• baggrunden is the definite form of baggrund
• together, the phrase means the background of the scene or picture

So:

• baggrund = background
• baggrunden = the background
• i baggrunden = in the background

Using på here would usually sound wrong for an ordinary scene description.

Because sin/sit/sine is a reflexive possessive, and it must refer back to the subject of the clause.

In this sentence, the subject is man.

So:

• sin/sit/sine would refer to man
• hendes refers to her, someone other than the subject

That is why hendes bøger is correct if the books belong to some woman or girl mentioned or understood from the context.

Compare:

• Hun ser sine bøger. = She sees her own books.
• Man kan se hendes bøger. = One can see her books.

Because possessives such as min, din, hans, hendes, vores already make the noun phrase definite enough.

So Danish normally says:

• hendes bog
• hendes bøger

not:

• hendes bogen
• hendes bøgerne

This is similar to English:

• her books not
• the her books

So the noun stays in its basic singular or plural form after the possessive.

In Danish, when a noun is definite and has an adjective before it, you normally use a separate definite article such as den, det, or de, and the noun itself appears without the definite ending.

So:

• lampen = the lamp
• den blå lampe = the blue lamp

Not:

• den blå lampen in standard Danish

This is an important pattern:

• bilen = the car
• den røde bil = the red car

So in your sentence:

• den marks definiteness
• blå is the adjective
• lampe is the noun in its basic form

Because the adjective form used here is the correct one for this noun phrase.

The noun lampe is a common-gender noun:

• en lampe

With blå, the forms are typically:

• en blå lampe
• et blåt vindue
• den blå lampe
• det blå vindue
• de blå lamper

So in den blå lampe, blå is correct.

The form blåt would be used with a neuter singular noun in an indefinite phrase, for example:

• et blåt vindue

Because ved means by, next to, or near.

So den blå lampe ved vinduet means the blue lamp is located beside the window.

By contrast:

• i vinduet would mean in the window, for example in the window space, on the windowsill, or in a shop window display

Also:

• vindue = window
• vinduet = the window

The ending -et is the definite singular ending for many neuter nouns:

• et vindue
• vinduet

So ved vinduet means by the window.