"Nobody believes that she has arrived by train."
Translation:Ingen tror på at hun er ankommet med tog.
I you take the following two sentences: "Jeg tror at hun snart ankommer" "Jeg tror på at hun snart ankommer" Both would translate to "I believe that she will arrive soon", but they have a slight but important different that is hard to translate to English.
"Jeg tror" means "I believe" as in "I am of the belief that" "Jeg tror på" means "I believe" as in "I believe it is true what someone told me that" In essence without 'på' you are just saying that you believe the statement to be true. With 'på' you you are saying that you believe someone else's statement to the following effect. It is a tricky very subtle difference that actually changes the meaning a lot.
So my eksample sentenses could maybe translate as following to carry the full meaning in English. "Jeg tror at hun snart ankommer" to"I am of the belief the she will arrive soon" "Jeg tror på at hun snart ankommer" to"I believe it is true what she told me that she will arrive soon" though the Danish sentence does not specify who told you and in order to translate this way you have to make that up.
I hope this helps.
And lastly for the original sentence the reason you have to use 'på' is because it would make no sense for you to just state that you don't think that she arrived be train without someone stating that she arrived be train. The most likely context would be that she said something about her train journey, but you for whatever reason don't actually believe that she arrived by train. But the sentence does not imply in anyway who made the statement that she arrived by train, though someone di.. Though since nobody believes it is it properly her who stated it.