Genvejen til stationen går gennem parken.

Because -en is the definite article attached to the noun. Danish usually puts the at the end of the noun instead of using a separate word.

• en genvej = a shortcut
• genvejen = the shortcut

So Genvejen til stationen means The shortcut to the station.

Genvej means shortcut. It is a compound noun:

• gen- = a prefix suggesting something like direct, quick, or shortened
• vej = road / way / path

Danish forms compound nouns very freely, much like English does. So genvej is literally something like short way or direct way.

Because til means to and shows direction or destination.

• genvejen til stationen = the shortcut to the station

This is the normal preposition here. You are describing where the shortcut leads.

For the same reason as genvejen: it is definite.

• en station = a station
• stationen = the station

So til stationen means to the station, not just to a station.

Because the sentence refers to the park, a specific park, not just any park.

• en park = a park
• parken = the park

So gennem parken means through the park.

In Danish, just like in English, routes, roads, paths, and similar things can be described as if they go somewhere.

So:

• Vejen går gennem skoven = The road goes through the forest
• Stien går langs floden = The path goes along the river
• Genvejen ... går gennem parken = The shortcut ... goes through the park

This is completely natural Danish.

Danish is a V2 language, which means the finite verb normally comes in the second main slot in a main clause.

Here the first element is:

• Genvejen til stationen

Then comes the verb:

• går

Then the rest:

• gennem parken

So the pattern is:

• [Subject] + [Verb] + [Rest]

This sentence is very standard Danish word order.

Yes, but the verb still has to stay in second position in a main clause.

For example:

• Gennem parken går genvejen til stationen.

This is grammatical, but it sounds more marked or literary, because gennem parken is put first for emphasis.

The most neutral version is:

• Genvejen til stationen går gennem parken.

Gennem means through.

It is used when something passes from one side of something to the other, or moves within an area:

• gå gennem byen = go through the city
• køre gennem tunnelen = drive through the tunnel
• gå gennem parken = go through the park

In this sentence, it tells you the route of the shortcut.

Yes, but they often overlap.

• gennem is the basic and most neutral word for through
• igennem can sometimes feel a bit more emphatic, like all the way through

In a sentence like this, gennem is the most natural choice:

• Genvejen til stationen går gennem parken.

Using igennem here would not usually be the best default choice.

Because Danish usually expresses definiteness with a suffix attached to the noun.

Compare:

• English: the station

• Danish: stationen

• English: the park
• Danish: parken

This is one of the most important differences between English and Danish noun grammar.

It would be:

• En genvej til en station går gennem en park.

But that sounds unnatural in most real contexts, because it means something like A shortcut to a station goes through a park, which is very vague.

The definite form is more natural when the speaker has a specific station and park in mind:

• Genvejen til stationen går gennem parken.

A rough guide is:

• genvejen ≈ GEN-vy-en

A few helpful notes:

• the stress is mainly on GEN-
• vej sounds roughly like English vy in Ivy, though not exactly
• the final -en is a weak syllable

The exact Danish pronunciation is difficult to represent in English spelling, but the important thing is to avoid pronouncing vej like English way too strongly.

It is attached to genvejen.

So the structure is:

• Genvejen til stationen = the shortcut to the station
• går gennem parken = goes through the park

In other words, til stationen describes which shortcut, and gennem parken describes where it goes.

Yes. It shows a very useful pattern:

• [definite noun] + til + [definite noun] + går + gennem + [definite noun]

For example:

• Vejen til stranden går gennem skoven.
  The road to the beach goes through the forest.

• Stien til søen går gennem marken.
  The path to the lake goes through the field.

This is a common and natural way to describe routes in Danish.