The modal verb "mogen" means "to be allowed to". This can be combined with the preposition "van" to indicate who is allowing this. The structure is hard to translate into a natural English sentence, especially with the negation. Hopefully this will help:
- "Hij mag fietsen." = He is allowed to bike.
- "Hij mag fietsen van zijn moeder." = He is allowed, by his mother, to bike. = His mother allows him to bike.
- "Hij mag niet fietsen van zijn moeder." = His mother does not allow him to bike.
Would a more literal (syntactically) translation be: "He may not bicycle, according to his mother"?
The way you phrase it changes the meaning: according to his mother, he is not allowed to bike. I.e. the mother tells us that he is not allowed to bike, but we don't know who forbade it.
There is a difference between literal translation and contextual translation. While "fietsen" by itself can mean "bicycle", in the context of the sentence it simply means "bike".