Paul wants to tell us that the life of the world to come is at work in the Christian already in the here and now. It was because of what Christ did that Paul could now experience eternal life. The spiritual life God has given doesn't take away all our pain and suffering, but despite our weaknesses God uses us to teach others about Him. Jesus explained to Nicodemus in John chapter 3 how that works. Paul's answer is Christ's death and resurrection, because that gives us hope.
Hope involves trust, committing our lives to an expectation.
As Christians our hope is set on nothing less than Christ alone. Because we have believed, we want to pass on the message to a world that is perishing. One of the best pictures of Christ's death and resurrection and the hope that it gives us is described in John Bunyan's book Pilgrim's Progress.