The first thing I have to say, just to gain some perspective, is that although the Church has been praying the Psalms from its very beginning, those who are themselves just beginning to pray the Psalms need to allow themselves several years to really start to get some verses and even some whole psalms. Instead of giving you an essay on the spiritual reading of Scripture, let me just share with you what I wrote to my questioning friend.
At the time this psalm was written, about eight hundred years before Christ, the only way to worship God was through actual animal sacrifices, of which part was burnt as an offering to God, part was eaten by the priest, and part was eaten by the person(s) who offered the animal for sacrifice. This was how God was worshiped in the Old Testament. In the New Testament, animal sacrifices are no longer offered, for Christ himself is the final sacrifice. However, every Christian offers him or herself as a living sacrifice, or as St. Paul says, as a drink offering poured out. Psalm 50 (51) is important because it makes clear that the real sacrifice is in the heart ("a broken and contrite heart O God You will not despise"). External acts reveal the heart, but without the humbled and broken heart, words and even actions of giving oneself to God mean nothing. So, when God "renews a right spirit within me" then I "teach transgressors the way, and the impious are converted...then will I offer bullocks [that is, my life in Christ] upon Your Altar."
So in praying Psalm 50 (51), one of the ways to spiritually interpret it is that I, in imitation of Christ, also am a sacrifice offered to God, a living sacrifice. Consequently, the mention of offering bullocks upon God's Altar is a reference to my own life offered to God. This is the fruit of the broken and contrite heart.