St. Athanasios the Great explains that we can see ourselves in some of the Psalms:
Those who listen (to other books) consider themselves to be other than those about whom the passage speaks, so that they only come to the imitation of the deeds that are told to the extent that they marvel at them and desire to emulate them. By contrast, however, he who takes up this book—the Psalter—goes through the prophecies about the Savior, as is customary in the other Scriptures, with admiration and adoration, but the other psalms he recognizes as being his own words. And the one who hears is deeply moved, as though he himself were speaking, and is affected by the words of the songs, as if they were his own songs.