After Peter received the revelation that Jesus is the Messiah, Jesus asked the disciples to not make this known to anybody. Why did he do this? We know that in other instances he did not do this, for instance the Samaritan woman in John 4. What made this case different?
John Dominic Crossan says in The Birth of Christianity, that a massive majority of scholars believe that much of the Gospel of Luke was based on Mark's Gospel, so we can say that Luke 9:21 requests silence because this verse was copied from Mark 8:30.
Mark 8:30 forms part of the theme now known by scholars as the 'Messianic Secret', by which Jesus is portrayed as commanding his followers to be silent about his Messianic mission. Throughout the Gospel of Mark, there are regular examples of Jesus admonishing both people and demons not to reveal to others who he truly was.
Holman Bible Dictionary suggests that Jesus commanded silence because of people's expectations or so that he could move around freely. However, this does not explain why the Messianic Secret is largely confined to Mark's Gospel and derivative material, nor does it explain why there are some passages in which Jesus actually told people to tell others about him, such as happens in Mark 5:19. It may be that, at the time Mark's Gospel was written, that it would have been dangerous for Christians to be associated with claims of messiahship.