Jeg vil gerne læse dette digt højt.

Together, vil gerne usually means would like to.

• vil on its own can mean want to, intend to, or sometimes function like will
• gerne literally means something like gladly or with pleasure

When you put them together, the result is often softer and more polite than plain vil.

So:

• Jeg vil gerne læse dette digt højt = I would like to read this poem aloud
• Jeg vil læse dette digt højt sounds more direct: I want to / I am going to read this poem aloud

Because that is the normal Danish word order here.

In a main clause like this, the finite verb usually comes early, and adverbs such as gerne often come before the infinitive.

So the pattern is:

• Jeg = subject
• vil = finite verb
• gerne = adverb
• læse = infinitive

That is why Jeg vil gerne læse ... sounds natural.

For an English speaker, it may help to think of gerne as sitting in the middle of the verbal expression:

• vil gerne læse = would like to read

Because læse is the infinitive, and after vil Danish uses the infinitive.

Compare:

• Jeg læser = I am reading / I read
• Jeg vil læse = I want to read / I will read

This is the same idea as in English:

• I read
• I want to read

After a modal verb like vil, kan, skal, må, and bør, Danish normally uses the bare infinitive.

Because after a modal verb like vil, Danish does not use at.

So you say:

• Jeg vil læse
• Jeg kan læse
• Jeg skal læse

not:

• Jeg vil at læse

This is similar to English, where we say:

• I can read
• I will read

not:

• I can to read

If you remove gerne, the sentence becomes more direct and stronger.

Compare:

• Jeg vil gerne læse dette digt højt = I would like to read this poem aloud
• Jeg vil læse dette digt højt = I want to read / I will read this poem aloud

So gerne often adds politeness, softness, or friendliness.

In many situations, English would like to is the best translation of vil gerne.

Because digt is a neuter noun, an et-word.

In Danish:

• denne is used with common gender nouns
• dette is used with neuter nouns

Examples:

• denne bog = this book
  because bog is an en-word
• dette digt = this poem
  because digt is an et-word

So the form of this changes to match the gender of the noun.

Because after denne/dette/disse, Danish normally uses the noun without the definite ending.

So:

• et digt = a poem
• digtet = the poem
• dette digt = this poem

Even though this poem is definite in meaning, Danish does not usually add the definite suffix here.

That is a very common thing for English speakers to notice, because English has this poem, while Danish uses:

• dette digt not
• dette digtet

Here højt means aloud or out loud.

So:

• læse højt = read aloud

Even though høj/højt often relates to high or loud, in this expression the natural English meaning is aloud, not just loudly.

So the sentence means that the speaker wants to read the poem so it can be heard, rather than reading silently.

Because højt is the -t form of høj, and in this sentence it functions adverbially.

Very roughly:

• høj = adjective form
• højt = neuter form, and also often the form used adverbially

This happens in many Danish words:

• hurtig → hurtigt
• smuk → smukt
• høj → højt

In læse højt, the -t form is the standard one.

Because that is the normal position here.

In læse dette digt højt, the object dette digt comes before højt, and højt finishes the verbal idea read aloud.

So the structure is:

• læse = read
• dette digt = this poem
• højt = aloud

This word order is natural in Danish. English also often puts aloud late in the sentence:

• read this poem aloud

So in that respect, Danish and English are quite similar here.

Dette digt is correct and natural, but it can sound a little more written or formal than everyday spoken Danish.

In casual speech, many Danes would often say:

• det her digt

So both are possible:

• Jeg vil gerne læse dette digt højt
• Jeg vil gerne læse det her digt højt

Very roughly:

• dette digt = slightly more formal, written, or careful
• det her digt = very common in everyday speech

They are very close, but not always identical.

• læse højt means read aloud / read out loud
• læse op also means read aloud, often especially to other people or from a text

In many situations, they can overlap.

Examples:

• Jeg vil gerne læse dette digt højt = I want to read this poem aloud
• Jeg vil gerne læse dette digt op = I want to read this poem aloud / read it out

A small difference in feel:

• læse højt emphasizes that it is not silent
• læse op often emphasizes reading something out to an audience or listener

Both are good Danish, but læse højt is perfectly natural in your sentence.