Min datter synes, at løbebåndet ser sjovt ud, men hun er for ung til at bruge det alene.

Because min already makes the noun definite in this kind of phrase.

In Danish, when you use a possessive like min, din, hans, hendes, vores etc., the noun normally appears in the indefinite form:

• min datter = my daughter
• hans bil = his car
• vores hus = our house

So min datter is the correct form, not min datteren.

The ending -et marks the noun as definite singular for a neuter noun.

• et løbebånd = a treadmill
• løbebåndet = the treadmill

Danish often adds the definite article to the end of the noun instead of using a separate word like English the.

Since løbebånd is a neuter noun, its definite singular form is løbebåndet.

Synes means something like thinks / feels / finds in the sense of giving an opinion.

So:

• Min datter synes, at ... = My daughter thinks that ... / feels that ...

It is very common in Danish for opinions:

• Jeg synes, det er godt. = I think it is good.
• Hun synes, filmen er kedelig. = She thinks the movie is boring.

It is not quite the same as English think in every context, but in sentences about personal opinion, synes is extremely common.

Here at means that and introduces a subordinate clause.

• Min datter synes, at løbebåndet ser sjovt ud
  = My daughter thinks that the treadmill looks funny/fun.

In English, that is often optional:

• My daughter thinks the treadmill looks fun.

In Danish, at is very commonly included in this structure.

Do not confuse this at with the infinitive marker to in phrases like til at bruge. Danish at can mean either that or mark an infinitive, depending on the structure.

Because ser ... ud means looks ... rather than is ....

• ser sjovt ud = looks funny / looks fun
• er sjovt = is funny / is fun

That is an important difference. The sentence says the treadmill appears amusing to her, not necessarily that it actually is.

This pattern is very common in Danish:

• Det ser godt ud. = It looks good.
• Det ser mærkeligt ud. = It looks strange.
• Hun ser træt ud. = She looks tired.

Because the adjective agrees with the noun løbebåndet, which is a neuter singular noun.

The base adjective is sjov.
When used with a neuter singular noun, it usually becomes sjovt.

• en sjov film = a funny movie
• et sjovt spil = a funny/fun game

Since løbebånd is an et-word, Danish uses:

• løbebåndet ser sjovt ud

So the -t matches the neuter noun.

Because se ud is a fixed verb expression, much like a phrasal verb in English.

• se ud = look / appear

In main clauses, Danish often separates the main verb and the particle:

• Det ser godt ud.
• Hun ser træt ud.

So ser and ud belong together in meaning, even though they are split in the sentence.

Those commas follow normal Danish punctuation patterns.

1. synes, at ...
   A comma often comes before a subordinate clause introduced by at.

2. ..., men ...
   A comma is used before the coordinating conjunction men = but.

So the commas help show the sentence structure:

• main clause: Min datter synes
• subordinate clause: at løbebåndet ser sjovt ud
• new coordinated clause: men hun er for ung til at bruge det alene

This is the standard Danish pattern for too ... to ...:

• for + adjective + til at + infinitive

So:

• for ung til at bruge det alene = too young to use it alone

More examples:

• for træt til at arbejde = too tired to work
• for lille til at nå = too small to reach
• for dyr til at købe = too expensive to buy

This is a very useful structure to memorize.

This at is different from the earlier one.

In til at bruge, at is the infinitive marker, like English to:

• at bruge = to use

So the sentence contains two different uses of at:

• synes, at ... = thinks that ...
• til at bruge = to use

They look the same, but their jobs are different.

Because det is the object pronoun referring back to løbebåndet.

• løbebåndet is a neuter noun
• the matching pronoun is det = it

So:

• bruge det alene = use it alone

Repeating the noun would sound unnecessary:

• ... til at bruge løbebåndet alene is possible, but less natural here.

Using det is exactly what English does with it.

Alene means alone / by herself / on her own.

In this sentence:

• bruge det alene = use it alone / use it by herself

It comes at the end because that is a very natural position in Danish for this kind of adverb.

Compare:

• Han bor alene. = He lives alone.
• Barnet må ikke gå derhen alene. = The child must not go there alone.

Here alene adds an important safety-related detail: she is too young to use the treadmill by herself.

Because men simply joins another main clause, and the normal main-clause word order is kept:

• hun er for ung ...

Danish only switches to verb-first order in questions or when another element comes first and triggers inversion.

Compare:

• Hun er for ung. = She is too young.
• Men hun er for ung. = But she is too young.
• I dag er hun for ung. = Today she is too young.

In the last example, I dag comes first, so the verb moves before the subject. But after men, nothing has been fronted inside the clause, so hun er stays in normal order.

Either can work depending on context, which is why sjov can be tricky for English speakers.

• sjov / sjovt can mean fun, funny, or amusing

In this sentence, ser sjovt ud could mean:

• looks fun
• looks funny
• looks amusing

A treadmill in this context probably looks fun to the daughter, but the Danish wording leaves a little flexibility. That is normal with sjovt.