Although Peter messed up a lot, Jesus still cared for Peter and looked ahead. He reinstated Peter and gave Peter new tasks, like feeding and tending sheep (AKA believers). But Jesus also showed Peter an outline of what Peter's new life should look like.
In the description, Jesus made a comparison. He talked about how young Peter "girded [himself] and walked where [he] wished", but old Peter "will stretch out [his] hands, and another will gird [him] and carry [him]" -- even in areas where Peter was hesitant on (21:18). And then Jesus told Peter, "Follow me." (21:19) In summary, Peter's next step is self-sacrifice. The new life involves not following Peter's own desires but following Jesus's desires (even in areas where Peter at first feels ippy). After all, stretching out one's hands conveys a sense of dependence.
Jesus's statements were "signifying by what death [Peter] would glorify God." In the Bible, "dying to oneself" means putting Jesus above ourselves, and that act would definitely glorify God. Peter's heart by then is grown and mature, and Jesus knows that Peter would focus on God more than himself. In fact, Peter took it so seriously he eventually became a martyr.