St. Teresa of Avila says that this prayer is clearly seen through the deep peace, contentment and calm that one experiences (The Book of Her Life, Ch 15:5, p. 141).
A French Jesuit, Jean-Pierre de Caussade (1675-1751) wrote Abandonment to Divine Providence. This book describes how a person should live his life doing the will of God. He abandons himself to God so that no matter what happens to him, he reacts, as best he can, in the way God wishes. An experienced soul, in time of trouble, "will remain quite unmoved by all turmoil, indifferent to all noise and uproar. Ignoring them all, it will stay peaceful and unmoved in that intimate peace where it benefits so greatly by the manifestation of its love. This is the center where it rests."
Paul wrote to the Philippians, "Always be joyful, then, in the Lord; I repeat, be joyful" (Phil 4:4). He continues telling the Philippians, "Never worry about anything; but tell God all your desires of every kind in prayer and petition shot through with gratitude; and the peace of God which is beyond our understanding will guard your hearts and your thoughts in Christ Jesus" (Phil. 4:7).
Jean-Pierre De Caussade, Abandonment to Divine Providence. Translated by John Beevers (New York: Image Books, 1975) 88.
At Monroe prison, a prisoner was bringing several others into the RCIA class. He was telling them of miracles in his life and inviting them to "come and see". I asked him to tell me one of the miracles that happened to him and he said that his world has fallen apart, but he remains at so much peace inside him, that he knows that this peace is from God.