Violinens lyd er lysere end trommernes, men begge dele passer godt til den samme melodi.

Violinens is the genitive form, meaning the violin’s.

• violin = violin
• violinen = the violin
• violinens = the violin’s

So violinens lyd literally means the violin’s sound.

In Danish, possession is often made by adding -s to the noun, much like English ’s.

Trommernes is also a genitive form, but this time from a definite plural noun.

• tromme = drum
• trommer = drums
• trommerne = the drums
• trommernes = the drums’

So trommernes means the drums’.

Because Danish, like English, often leaves out a repeated noun when it is obvious from context.

So:

• Violinens lyd er lysere end trommernes
  means
• The violin’s sound is brighter than the drums’ [sound]

The word lyd is understood the second time, so it does not need to be repeated.

Lysere is the comparative form of lys.

• lys = light / bright
• lysere = lighter / brighter

Here it describes sound, so it means something like brighter in tone.

Danish often uses lys and mørk not only for color, but also for sound:

• en lys lyd = a bright/light sound
• en mørk lyd = a dark/deep sound

End means than after a comparison.

So:

• lysere end trommernes = brighter than the drums’

This is the normal Danish pattern:

• større end = bigger than
• bedre end = better than
• hurtigere end = faster than

Begge dele means both things or simply both when talking about two non-personal things or ideas.

In this sentence, it refers to the two things just mentioned: the violin’s sound and the drums’ sound.

Why not just begge?
You sometimes can say just begge, but begge dele is very common when the meaning is both of those things/aspects.

Compare:

• Begge kom i går = Both came yesterday.
  Used for people or clearly countable subjects.
• Jeg kan lide begge dele = I like both.
  Used for two things/options.

Passer til means fits, suits, or goes well with.

So:

• passer godt til = fits well with / goes well with

In the sentence:

• begge dele passer godt til den samme melodi
  = both go well with the same melody

The word godt is an adverb here, meaning well.

This means the same melody.

• den = the / that (common gender form here)
• samme = same
• melodi = melody

Melodi is a common gender noun, so it takes den, not det.

This pattern is very common in Danish:

• den samme bog = the same book
• det samme problem = the same problem
• de samme sange = the same songs

Because men means but and links two main clauses.

The two clauses are:

• Violinens lyd er lysere end trommernes
• men begge dele passer godt til den samme melodi

In standard Danish writing, a comma is normally used before men when it joins two full clauses.

Grammatically, it most directly refers to the two things being compared in the sentence, which are the sounds:

• the violin’s sound
• the drums’ sound

But in natural interpretation, it can also feel like it refers more broadly to the two instruments as musical options. The immediate grammar points to the sounds, while the overall meaning can include the instruments themselves.

Yes. After men, you start a new main clause, and the word order here is completely normal:

• begge dele = subject
• passer = verb
• godt til den samme melodi = rest of the clause

So it follows the basic main-clause pattern:

• subject + verb + other elements

That is why begge dele passer sounds natural.