I dag lærer vi at bruge “den her”, “det her” og “de her”, når vi peger på noget tæt på.

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

When I dag comes first, it takes the first position in the sentence, so the finite verb must come next:

• I dag lærer vi ...
• Compare: Vi lærer i dag ...

So lærer comes before vi because I dag is already in first place.

Here it means learn.

The verb lære can sometimes correspond to both learn and teach, depending on the structure:

• Vi lærer at bruge ... = We are learning to use ...
• Hun lærer børnene dansk = She teaches the children Danish

So in your sentence, because it is lærer ... at bruge, the meaning is clearly learn.

At is the infinitive marker, like English to.

So:

• bruge = use
• at bruge = to use

After lære, Danish often uses at + infinitive:

• lære at læse = learn to read
• lære at skrive = learn to write
• lære at bruge = learn to use

They match the gender and number of the noun:

• den her = singular common gender
• det her = singular neuter
• de her = plural

Examples:

• den her bog = this book
• det her hus = this house
• de her bøger = these books

So you choose the form based on the noun that follows, or on the thing the phrase refers to.

Normally, no.

You say:

• den her bog
• det her hus
• de her bøger

not:

• den her bogen
• det her huset
• de her bøgerne

In standard Danish, den her / det her / de her already work as demonstrative determiners, so the noun usually stays in its basic form.

Yes. They can stand on their own.

Examples:

• Hvad er det her? = What is this?
• Jeg tager den her. = I’ll take this one.
• De her er mine. = These are mine.

So they can be used either:

• with a noun: den her stol
• without a noun: den her

Yes, but the tone is a little different.

• den her / det her / de her are very common in everyday spoken Danish
• denne / dette / disse are more formal, written, or bookish

Examples:

• den her bog = very natural in speech
• denne bog = correct, but more formal

Both are right, but for normal conversation, learners will hear den her / det her / de her very often.

That is the normal Danish pattern for these common demonstratives:

• den her
• det her
• de her

Here, her means here, and together the phrase means something like this/these here, meaning something close to the speaker.

There is also a contrast with der for something farther away:

• den her = this one here
• den der = that one there

So the order is fixed: den her, not her den.

Because they belong to two different expressions:

• pege på = to point at
• tæt på = close to / nearby

So in:

når vi peger på noget tæt på

the first på goes with peger, and the second på goes with tæt.

It may look repetitive, but grammatically it is completely normal.

Because når means when / whenever, while hvis means if.

In this sentence, the idea is general: this is what we use when we point at something nearby. It is not really a condition; it is a usual situation.

• når vi peger på noget tæt på = when / whenever we point at something close by
• hvis vi peger på noget tæt på = if we point at something close by

So når is the natural choice here.

Because noget means something and is used for an unspecified thing.

Here, the sentence is talking about pointing at something nearby, not someone.

Compare:

• noget = something
• nogen = someone / anyone / some in other contexts

Examples:

• Jeg ser noget. = I see something.
• Jeg ser nogen. = I see someone / some people.

So noget is correct because the sentence refers to an unspecified thing.

Because the sentence is talking about the expressions themselves.

It is not using them to point at actual objects. Instead, it is naming the forms being taught. In English, we do the same thing when we mention words as words.

So the quotation marks show that these are language items under discussion.