Those who hunger and thirst for righteousness are those people who deeply acknowledge their sinfulness, their guilt before God, and have a burning desire for righteousness. They try to serve God by a righteous life according to the commandments of Christ, which requires from Christians the most holy righteousness in all their relations with their neighbors.
The expression "hunger and thirst" indicates that our yearning for righteousness must be very strong, as strong as our desire to appease our appetite and thirst. King David beautifully expressed such yearning, As the hart panteth after the fountains of water, so panteth my soul after thee, O God. My soul thirsted for God, the mighty the living (Ps. 4:1-2).
God promised that those who hunger and thirst for righteousness will be filled. By this is meant spiritual satisfaction, comprised of internal spiritual peace, a calm conscience, justification, and forgiveness. Such satisfaction in the present, earthly life occurs only in part. The Lord reveals the mysteries of His kingdom to those who hunger and thirst after righteousness, more than to others. Their hearts in this world are delighted with knowledge revealed in the divine truths of the Gospel, in Orthodox teachings.
Full satiety, full satisfaction of the holy yearnings of the human soul, and from this highest joy and blessedness, will be granted them in the future, blessed life with God. As the psalmist King David says, I shall be filled when Thy glory is made manifest to me (Ps. 16:16).