Translation:Here you can see the arrivals and departures of trains.
Do you have any reference for your claim. It seems to be normal https://books.google.com/ngrams/graph?content=Here%20you%20can%20see
As a native English (American) speaker, the sentence "You can see the train arrivals and departures here" is correct and sounds more organic and ordinary. The sentence "Here you can see the arrivals and departures of the trains" is actually what I put for my translation, but I tend to be more formal and literary in my speech. The second sentence actually sounds to me more like British English than American, but both make perfect sense and are grammatically correct. I think most Americans would just say something like, "Here are the train arrivals and departures."
It sounds like something a tour guide would say, maybe followed by "and over there you can see a cafe". As opposed to a part of a conversation about how to take a train, like "you will need a ticket, and you can see the arrivals and departures here". Both correct, but the former seems more like part of a script.
"Train's" is a singular possessive. Here, it would be referring to the arrivals and departures of one train. But the Czech sentence contains vlaků, which is genitive plural, so we know that multiple trains are involved. It is possible that "trains'" (note that the apostrophe is AFTER the "s" in "trains") is also accepted.