To share the life-transforming hope of Jesus is to share Jesus himself (the person and promises of Christ). How can we share this hope?
The prayer in Romans 15:13 provides a starting point for how we can share the life-transforming hope of Jesus Christ. As we trust in the God of hope, he will fill us with all joy and peace so that we may overflow with hope by the power of the Holy Spirit.
We start with our actions but a watching world will have questions. When these questions come, we are called to respond to these questions with a reason for the hope that we have (1 Peter 3:15). God calls us to provide an explanatory word about God's work in our lives. We are called to live "questionable lives" (lives that are called into question for good). This is a different kind of questionable life than how we normally use that description.
In summary, as we revere (set aside) Christ as Lord in our hearts, the prayer of Romans 15:13 will be answered and take place. We should then ALWAYS be prepared to give an answer to everyone who asks. We can share the life-transforming hope of Jesus Christ in this way and invite others to (re)consider this hope in their lives.