Jesus is the Divine Physician (see Mt 9:12). In an operation to remove a malignant tumor, a surgeon divides the healthy cells from the cancerous cells. He removes only the cancer and leaves the rest. Likewise, Jesus divides the good and bad (Mt 25:32).
Jesus is "the Carpenter" (Mk 6:3). In His workshop, He divided strong wood from rotten wood. He divided broken nails from sturdy ones. "What was useless [He] threw away" (Mt 13:48).
Jesus is the Word of God (Jn 1:1). He, the Living Word, "divides soul and spirit, joints and marrow" and "judges the reflections and thoughts of the heart" (Heb 4:12). He divides us from sinful thoughts, bitterness, anger, unforgiveness, and impurity.
Jesus is Family with the Holy Spirit and "the Father from Whom every family in heaven and on earth takes its name" (Eph 3:14-15). As He divides all else, Jesus also divides families if they need to be rebuilt or restored (Lk 12:52-53). If necessary, Jesus will divide father and son, mother and daughter, in-laws, and anyone or anything that divides the family from God's plan.
Why has Jesus "come for division"? (Lk 12:51) It is because God is Love (1 Jn 4:8, 16). Love is consuming. Love suffers "anguish" (Lk 12:50) until we "attain to the fullness of God Himself" (Eph 3:19). Love can't tolerate anything holding us back from that fullness. Therefore, repent completely of anything that holds you back from the Lord — before Jesus has to divide you from it.