“Now while the Pharisees were gathered together, Jesus asked them a question, saying, “What do you think of the Christ? Whose son is he?” They said to him, “The son of David.” He said to them, “How is it then that David, inspired by the Spirit, calls him Lord, saying, `The Lord said to my Lord, Sit at my right hand, till I put thy enemies under thy feet’? If David thus calls him Lord, how is he his son?”
Let me give you the explanation right out of the Ignatius Catholic Study Bible…”Jesus’ question concerns the meaning of Ps 110:1. The Pharisees (22:42) assume it mentions the Messiah, but their understanding of the verse is partial and inadequate. Psalm 110 is an enthronement psalm that was probably used at coronation ceremonies for Davidic kings. In context, David addresses his son as “my Lord” (22:44), a title more appropriate for one’s superior. This implies that the expected Messiah would be greater than David himself, a crucial point missed by the Pharisees (cf Acts 2:34-36). As Messiah, Jesus is the son of David (Matt 1:1) and yet greater than David as the Son of God (Matt 3:17; 16:16; 17:5) (CCC 439, 447).”

The Mosaic Law instituted Stage Two of salvation history, the Age of Law. It began with Moses and lasted through the history of ancient Israel (see question “If because of Adam’s fall and his loss of Grace, were all people without Grace until Christ restored it?”). Through it, the Jews received both the moral law (i.e. 10 Commandments) and ceremonial law (“works of the law” in the writings of St. Paul).
The ceremonial laws were temporary and given so that the people would recognize their sinfulness and desire true righteousness. (This would finally come true through Christ.) They needed to recognize that they needed “circumcised hearts” not just fleshly circumcision (see Deuteronomy 30).
In other words, the law was given to show the need for the grace of the Spirit. It shows us what sin is. What then shall we say? That the law is sin? By no means! Yet, if it had not been for the law, I should not have known sin. I should not have known what it is to covet if the law had not said, “You shall not covet” (Romans 7:7).
The law (Torah) given to Moses by God was a great advantage for the covenant people compared to the rest of the world because it ordered their worship, gave them clear guidelines for living, and drew them closer to God. The law was a necessary step toward the progress of the coming of Jesus Christ. It helped prepare the way by showing people their need for new hearts.

It would be more correct to say that in the Old Testament people didn’t have the indwelling of the Trinity in the same manner we do now through the sacrament of baptism, but they still had an avenue to saving grace.
Just as Our Lady was saved by the application of what Christ merited on the cross before it actually happened in history, so too could those in the Old Testament receive the benefits of Christ’s sacrifice before it actually happened in time. This would be considered what the Catechism calls the Baptism of Desire (CCC 1260).
We would call this Stage One of salvation history in which people lived under the natural law. In fact, there are 3 stages of salvation history which we can infer from Romans 5:12-14, 19-21 (St. Thomas Aquinas also discusses them directly in his Commentary on St. John (n. 338. 150):

1. The age of nature, which begins at creation, runs through the era of the patriarchs, to the time of Moses
2. The age of law, beginning when Moses received the law from God at Mt. Sinai continuing through the history of ancient Israel; The Mosaic law, as it is called, contained a moral law which continues to be in effect (i.e. the 10 Commandments) and ceremonial laws which pertained only to Israel (e.g. circumcision). These ceremonial laws were superseded by the New Covenant law of grace which came through Jesus Christ.
3. The age of grace, which began with the advent of Jesus Christ. (This is also referred to as the age of the Church…cf CCC 1076.)

In his discussion of those abiding by the natural law before the time of Christ, St. Paul says that “When Gentiles who have not the law do by nature what the law requires, they are a law to themselves, even though they do not have the law. They show that what the law requires is written on their hearts, while their conscience also bears witness and their conflicting thoughts accuse or perhaps excuse them on that day when, according to my gospel, God judges the secrets of men by Christ Jesus” (Rom 2:14-16).
In other words, Gentiles follow the natural law that God has inscribed on their hearts. They didn’t have the privilege of the Mosaic law by birth (as did the Israelites). When Paul says that the Gentiles keep the law by nature, he means, not by nature apart from grace, but by nature that is healed and restored by grace.
A modern example of this would be how God can save an African whose tribe is unknown to the rest of the world and has not heard the Gospel. Though sacraments are now the ordinary means of salvation, we know that God is not bound by the sacraments and so can apply the merits of Christ as he wills.