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In short, these sacrifices reminded the Israelites of their imperfect relationship with God, and thus that they needed to seek and serve the Lord on a regular basis to draw closer to God instead of going further astray.
Thus, the Old Covenant sacrificial system served as a good teacher for the Israelites, who were reminded on a daily basis—as parents do in imparting lessons to their children—to seek and serve the Lord. In addition to daily sacrifices, there were key annual sacrifices like Passover (Ex. 12) and the Yom Kippur/Day of Atonement offerings (Lev. 16).
Yet even the Yom Kippur sacrifices could not provide perfect atonement, though they reminded the Israelite faithful to keep their eyes on God, who would one day send a Savior (see Is. 53) in establishing the New Covenant (Jer. 31:31-34). Jesus Christ is that Savior, who fulfills all Old Covenant sacrifices, including Yom Kippur (Heb. 9:11-14), in his one Sacrifice of Calvary, and thereby offers salvation to the whole world, Jew and Gentile alike (see CCC 846-48).