From Scripture Studies:
In the first five verses, David offered up his own blamelessness as a reason that God should answer his prayer. Here, David gives a second reason for God to answer his prayer, that is, God’s love for His people: “I call on You, O God, for You will answer me; give ear to me and hear my prayer. Show the wonder of Your great love, You who save by Your right hand those who take refuge in You from their foes.” David knows of God’s love through past experience with answered prayer. This is why he says confidently: “I call on You, O God, for You will answer me.” When you trust the Lord, it’s a never-ending, glorious cycle: dependence upon Him begets confidence, confidence leads to answered prayer, answered prayer leads to further dependence, etc.
David enumerates some of the aspects of God’s love for His people. God “save[s] by [His] right hand.” The right hand is the hand of strength, so David is saying that God does not pull punches when coming to the rescue of His people. David prays to God: “Keep me as the apple of Your eye; hide me in the shadow of Your wings…” The “apple of Your eye” refers to one’s pupil, which is a very valuable, yet small and easily injured, part of the body. Because of this, God has designed special protections for our pupils. They set into the socket of the eye so that they physically have special protections. God has also given us special reflexes which naturally protect the eye when danger threatens it. David is asking for the same sort of special protection.
David asks for similar protection when he asks to be hidden “in the shadow of [God's] wings.” This speaks of the special protection that a bird gives its chicks. Jesus desires to give His people such protection. In fact, He lamented that the people of Jerusalem did not “hide in the shadow of [His] wings”: “O Jerusalem, Jerusalem, you who kill the prophets and stone those sent to you, how often I have longed to gather your children together, as a hen gathers her chicks under her wings, but you were not willing.” (Matt. 23:37).