"The athlete does not run in onto the grass, but sits down onto the ground."
Translation:A sportoló nem fut be a fűre, hanem leül a földre.
Because the language we are trying to learn is Hungarian and not English. Sometimes there is no direct translation between the two, especially where the prefixes and suffixes of motion are concerned. So in order to show the real meaning of the Hungarian sentence, it is necessary to use an unnatural sentence in English. In this case they are trying to show that "befut" means "run in" and "fűre" means "onto the grass".
I'm not at all sure, so this is speculative, but comparing this sentence with:
"A gyerekek nem lesétálnak az utcára, hanem leugranak"
it seems that perhaps here (unlike in the linked sentence) we don't have a direct contrasting of two verbs, just a juxtaposition of two situations, so here the focus is on "nem" (like in most sentences with a negation) and hence the prefix is separated from the verb.
In any case, this is a fascinating (if, to me, slightly elusive) topic.