How is it that this is the case? In the first place it is because invoking the name of Jesus means invoking his very presence both within us and in the world immediately around us. By invoking Christ's presence within us, we ourselves become progressively more transformed into the Divine "likeness" that was lost due to sin. The Catechism of the Catholic Church, in paragraph 2713, points out that communion with God, attentive awareness of his presence in our hearts, "conforms man, the image of God, 'to his likeness.'" As this transformation progresses our actions become the very actions of God through us in the world. In speaking of man's dependence on the grace of God, St. Theophan the Recluse says:
"When making a vow to live according to the will of God, for His glory, the penitent should say, 'Only do Thou strengthen me and confirm my resolve'; and from then on he must, as it were, place himself every minute in the hand of God, with the prayer, 'Do Thou Thyself perform within me what is pleasing to Thy will'. In this way, alike in consciousness and will and in actual fact, it is God who will be acting in us, both in what we desire and in what we do according to His good pleasure" (taken from The Art of Prayer: An Orthodox Anthology, pg. 134: emphasis mine).
So our loving attentiveness to the Beloved within us, an attentiveness begun and strengthened by our prayerful repetition of the Jesus Prayer, conforms us to God's likeness and then prompts us to go out and be "the arms, hands, legs, feet, and voice of Christ in the world," to paraphrase St. Teresa of Avila. Fr. George Maloney, a Russian Greek/Byzantine Catholic priest, says:
"Jesus wishes to transfigure the whole world through us, by our humble actions in the world... By pronouncing the holy name of Jesus we release this transfiguring power. We call him into being to touch a world groaning in travail. We ask him to transform the universe, to make each human being, each part of God's creation into members of the Body of the Risen Lord. In such prayer, redemption is experienced as the process by which Jesus Lord transfigures the world through other loving human beings who allow him to have his redeeming way in them" (taken from Prayer of the Heart pg. 142).
This brings us to the second point, mentioned by Archbishop Raya in the Byzantine Daily Worship. All prayer, whether it's the private recitation of the Jesus Prayer, the Rule of the Theotokos, the rosary, etc., or the public worship of the Church in the Eucharistic Liturgy and the Liturgy of the Hours, makes the Kingdom of God present in this world. All created things are contained in this Kingdom because ultimately all things come from God and return to God. The cry Kyrie Eleison, "Lord, have mercy," introduces into this world the transforming presence of God into the world by making that presence conscious, respected and loved by those who repeat the prayer. The cry for mercy is not to be understood in a juridical manner, but in this manner of loving, transformative presence. It is a cry invoked for all of creation, as well as for all past, present and future generations.
But I believe St. Seraphim of Sarov best summed up this teachings in his now famous phrase, "Acquire the Spirit of Peace and a thousand around you will be saved." Through the prayer of the heart, especially the Jesus Prayer, we ourselves are transformed and transfigured. Through this transfiguration Christ in us increases while we, with all of our self-centeredness and selfish desires and impulses, decrease. The world then looks at us and sees Christ. By seeing Christ in us, may all come to give glory to God.