The following are some reasons as to why we should evangelise

Because Jesus Christ commanded us, the Church, to do so.
The Church’s Mission is to be missionary. In what is called “the Great Commission” Jesus said to his disciples:

“All authority in heaven and on earth has been given to me. Go, therefore, and make disciples of all nations, baptising them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.”

Jesus commands this because “He desires all men to be saved and to come to the knowledge of the truth”. We are invited to cooperate with Jesus’ desire by sharing the Good News, and we can be confident that Jesus is with us when we do so.

To share what we have been given.
True evangelisation flows out of an encounter with the person of Jesus Christ, who loves us with an “everlasting love” , and offers us “life to the full” . When we have experienced the goodness of the life and infinite love that Christ offers, the thought that a loved one doesn’t know that love and goodness can sadden us. Evangelists often share the good news because they are moved by compassion for those who do not know Him. They act in the hope others might find that same “life to the full” in the love that Christ offers.

Everyone should have a chance to know Christ.
We have a great gift because we know Christ, but in this day and age, some people do not have that gift. Everyone should be given the chance of knowing Christ, so that they have the option of believing in Him. Because we know Him, we have a duty to others to share Christ with those who have not had a chance to know Him.

John Paul II: “The Church cannot fail to proclaim that Jesus came to reveal the face of God and to merit salvation for all humanity by his cross and resurrection.”

Because it is good to share good news, and when the news is this good, you just have to share it!

John Paul II: “The Church offers mankind the Gospel… which responds to the needs and aspirations of the human heart and always remains ‘Good News.’”

Christ has saved us from our sin, and if we understand the importance of this, we will willingly share Christ with others. How could we not want to!?
Pope Paul VI: “It is unthinkable that a person should accept the Word and give himself to the Kingdom without becoming a person who bears witness to it and proclaims it in his turn.”