Barnet bruger en gul farveblyant til sin tegning.

Because Danish usually makes a singular noun definite by adding the definite ending to the noun itself.

• et barn = a child
• barnet = the child

So barnet is the normal way to say the child.

det barn can exist, but it usually means something more like that child or the child (in contrast to another one), depending on context.

Because farveblyant is a common gender noun in Danish, so it takes:

• en in the indefinite singular
• adjectives in the common gender form

So:

• en farveblyant = a colored pencil / crayon
• en gul farveblyant = a yellow colored pencil

If the noun were neuter, you would use et and often the adjective would also change form.

Because gul is describing a common gender singular indefinite noun: en ... farveblyant.

In Danish, adjectives change depending on gender/number/definiteness. Here the basic form is used:

• en gul farveblyant

Compare:

• et gult hus = a yellow house
• gule huse = yellow houses
• den gule blyant = the yellow pencil

So gul is correct because farveblyant is common gender singular and indefinite.

Yes. Danish writes compound nouns as one word.

So:

• farve = color
• blyant = pencil
• farveblyant = colored pencil / coloring pencil / crayon depending on context

This is very common in Danish. English often uses separate words where Danish uses a compound.

Bruger is the present tense of at bruge = to use.

So:

• jeg bruger = I use / am using
• barnet bruger = the child uses / is using

In Danish, the present tense is often used where English might use either the simple present or the present continuous, depending on context.

Because this is a normal Danish main clause with the subject first:

• Barnet = subject
• bruger = verb
• en gul farveblyant = object
• til sin tegning = prepositional phrase

So the pattern is basically:

Subject + verb + object + other information

That is the most straightforward word order in Danish main clauses.

Because Danish uses the reflexive possessive sin/sit/sine when the owner is the subject of the clause.

Here, the subject is Barnet, and the drawing belongs to that same child. So Danish uses sin:

• Barnet bruger en gul farveblyant til sin tegning.

This means the drawing is the child's own drawing.

If you said hans or hendes, it would usually mean the drawing belongs to someone else, not the subject.

Because sin/sit/sine agrees with the thing possessed, not with the owner.

The possessed noun here is tegning (drawing), and tegning is a common gender singular noun. So the correct form is:

• sin tegning

Compare:

• sin bog = his/her own book
• sit hus = his/her own house
• sine bøger = his/her own books

So even though the subject is barnet (a neuter noun), the form is sin because tegning is common gender.

Here til means something like for.

• til sin tegning = for his/her own drawing

It shows what the colored pencil is being used for.

If you left it out, the sentence would still be grammatical:

• Barnet bruger en gul farveblyant.
  = The child is using a yellow colored pencil.

But with til sin tegning, you add the purpose or context: the pencil is being used for the drawing.

Sometimes, but the meaning changes a little.

• til sin tegning focuses on purpose: the pencil is used for the drawing
• på sin tegning focuses more on the surface/result: the child is using it on the drawing

So til is about what the pencil is for, while på is more physical and specific.

Because the possessive already makes the noun definite enough.

In Danish, just like in English, you normally do not use an article with a possessive:

• sin tegning = his/her own drawing
• not sin en tegning
• not sin den tegning

So once you have sin, you do not add en or the definite ending.

Yes. Barnet simply means the child and does not specify male or female.

That is one reason sin is useful here: it does not force you to choose his or her in the same way English often does.

A very rough English-friendly approximation is:

BAR-neð BROO-er en gool FAR-vuh-blyant til seen tie-ning

A few notes:

• d at the end of barnet is soft, not a strong English d
• bruger has a Danish vowel sound that does not match English exactly
• y in blyant is not like normal English y
• tegning has a soft Danish pronunciation that is hard to spell accurately in English

So this approximation is only a starting point. Listening to native audio is very important for Danish pronunciation.