Die Fahrgastrechte gelten auch nach einem Zugausfall.

Because Fahrgastrechte is plural.

• das Recht = the right
• die Rechte = the rights

In German, the plural definite article is die for all genders, so:

• das Fahrgastrecht = the passenger right / passenger entitlement (singular)
• die Fahrgastrechte = the passenger rights (plural)

The sentence is talking about passenger rights in general, so the plural is natural.

It is a compound noun:

• Fahrgast = passenger
• Rechte = rights

So Fahrgastrechte literally means passenger rights.

German often combines nouns into one longer word. The most important part is usually the last part, because that determines the basic meaning and grammatical gender/number. Here, the last part is Rechte, so the whole word is plural.

gelten means something like:

• to be valid
• to apply
• to remain in force

So Die Fahrgastrechte gelten means The passenger rights apply / remain valid.

This is a very common verb in official, legal, and informational German. You will often see it in sentences like:

• Die Regel gilt auch hier. = The rule also applies here.
• Das Ticket gilt bis Montag. = The ticket is valid until Monday.

Because the subject is plural:

• die Fahrgastrechte = plural

So the verb must also be plural:

• die Fahrgastrechte gelten

Compare:

• Das Fahrgastrecht gilt. = singular
• Die Fahrgastrechte gelten. = plural

auch means also or too.

In this sentence, it tells you that passenger rights apply in this situation as well. So the idea is:

• the rights apply in general
• and they apply also after a train cancellation

Here auch nach einem Zugausfall is best understood as also after a train cancellation or even after a train cancellation, depending on context.

Because nach takes the dative case when it means after.

So:

• nach
  • dative

The noun Zugausfall is masculine:

• der Zugausfall

Its indefinite article in the dative singular is:

• einem

So:

• nach einem Zugausfall = after a train cancellation

It is the same word, but here it has a different meaning.

nach can mean:

• to with cities/countries: nach Berlin
• after in time: nach dem Essen
• according to: nach dem Gesetz

In your sentence, it means after in a time/event sense:

• nach einem Zugausfall = after a train cancellation

So this is not about movement; it is about what happens following an event.

Because nach requires the dative, not the genitive.

• Dative: nach einem Zugausfall
• Genitive: wegen eines Zugausfalls = because of a train cancellation

So the form depends on the preposition:

• nach
  • dative
• wegen
  • genitive

That is why einem Zugausfall is correct here.

It is another compound noun:

• Zug = train
• Ausfall = failure, cancellation, disruption

So Zugausfall literally suggests something like train cancellation/failure, but in normal English train cancellation is the most natural translation.

German compounds are often more literal than English, so it helps to break them into parts.

Because this is a normal main clause, not a subordinate clause.

German main clauses usually have the finite verb in second position:

• Die Fahrgastrechte | gelten | auch nach einem Zugausfall

So gelten is already in the correct place. The phrase auch nach einem Zugausfall is just an adverbial phrase giving extra information.

A verb at the end would be expected in a subordinate clause, for example:

• ..., weil die Fahrgastrechte auch nach einem Zugausfall gelten.

Yes. German word order is flexible as long as the finite verb stays in second position in a main clause.

For example:

• Auch nach einem Zugausfall gelten die Fahrgastrechte.

This version puts more emphasis on after a train cancellation.

Both are correct, but the original version is very natural in neutral informational German.

They are similar, but not always used in exactly the same way.

• Fahrgast is very common for someone using public transport such as a bus or train.
• Passagier also means passenger, but can sound broader and is often used for planes, ships, or travel generally.

In railway or public transport contexts, Fahrgastrechte is the standard term for passenger rights.