The doctrine of a "broken heart and a contrite spirit" begins with the law of sacrifice as contained in the Old Testament. Under that law, the firstborn were sacrificed in similitude of the Only-Begotten Son. We are quick to teach that the law of animal sacrifice was fulfilled in the ultimate sacrifice of Jesus Christ. We are sometimes slow to realize that there was a replacement "law of sacrifice." The new law, as taught by the Savior, requires just as regular and frequent a sacrifice. The difference is that we are to sacrifice a broken heart and a contrite spirit upon the altar of discipleship.
Through our faith in Jesus Christ we become his disciples and offer up a broken heart and contrite spirit. It is through this faith and sacrifice that we can obtain forgiveness of sins, for sanctification cometh because of their yielding their hearts unto God (Hel 3:35). Behold, he offereth himself a sacrifice for sin, to answer the ends of the law, unto all those who have a broken heart and a contrite spirit; and unto none else can the ends of the law be answered (2 Nephi 2:7). This doctrine is even contained in the Old Testament, For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering. The sacrifices of God are a broken spirit: a broken and a contrite heart (Ps 51:16-17).
"…in the scriptural usage a broken heart is a malleable, meltable, moveable heart, and a contrite spirit is an honest, acknowledging spirit that says, 'I am, in fact, dependent…' There is not self-deprecation here, only honesty: 'I need help.' And when that is acknowledged, help comes." (Truman Madsen, The Radiant Life, p. 113)
Ezra Taft Benson
"Godly sorrow is a gift of the Spirit. It is a deep realization that our actions have offended our Father and our God. It is the sharp and keen awareness that our behavior caused the Savior, He who knew no sin, even the greatest of all, to endure agony and suffering. Our sins caused Him to bleed at every pore. This very real mental and spiritual anguish is what the scriptures refer to as having 'a broken heart and a contrite spirit.' (See 3 Ne. 9:20; Moro. 6:2; D&C 20:37, 59:8; Ps. 34:18; Ps. 51:17; Isa. 57:15.) Such a spirit is the absolute prerequisite for true repentance." (Ensign, Oct. 1989, p. 2)
Neal A. Maxwell
"The real act of personal sacrifice is not now nor ever has been placing an animal on the altar. Instead, it is a willingness to put the animal that is in us upon the altar—then willingly watching it be consumed! Such is the 'sacrifice unto [the Lord of] a broken heart and a contrite spirit.' (3 Nephi 9:20.)" (Meek and Lowly, p. 94)
Bruce C. Hafen
"When we accept the modern law of sacrifice through the two-way covenants of the Atonement, our willingness to give the Lord everything we have mirrors his willingness to give us everything he has—our broken heart for his broken heart. The combination of his sacrifice with ours in the miraculous chemistry of this mutual belonging then gives us the power 'to contend against all the opposition, tribulations, and afflictions which [we] will have to encounter in order to be heirs of God, and joint-heirs with Christ Jesus.'" (Belonging Heart, p. 157)