Catholic understanding of sanctifying grace, from Divine Intimacy by Father Gabriel of St. Mary Magdalen, OCD, 42:
"Jesus tells us to abide in Him because we have been grafted on to Him. This spiritual engrafting, an accomplished fact, was made possible for all men by Christ's death on the Cross, and it became effective for each one of us at the time of our Baptism. Christ grafted us into Himself, at the cost of His precious Blood [and pure Body]. Therefore we are in Him, [by sanctifying grace], but He insists further that we abide in him and bring forth fruit.
Baptism is sufficient to graft us into Christ, and [the smallest] degree of grace will permit us to abide in Him like living branches, but we should not be content with this union only. We must show our gratitude for the immense gift we have received by endeavoring to become more and more firmly grafted into Christ. We must live with this union with Christ, making Him the center, the sun of our interior life.
"Abide in me" is not a chance expression. Christ wished to show us that our life in Him requires our personal collaboration with Him, that we are to employ all our strength and mind and will and our heart that we may live in Him and by Him. The more we try to abide in Christ the deeper our little branch will grow into Him, because it will be nourished more abundantly by the sap of grace.
"Abide in me, and I in you." The more closely we are united to Christ by faith, charity and good works done with the intention of pleasing God, the more intensely will He live in us and bestow on us continually a new life of grace. Thus we shall become not merely living branches but branches laden with fruit, the fruit of sanctity destined to bring joy to the Heart of God, for Jesus has said: "In this is My Father glorified, that you bring forth very much fruit [John 15:8]."