How is the matter of why the Son of God needed to be sacrificed explained in the Roman Catholic Church?

Starting with some definitions

What is "sacrifice"?

A sacrifice is the offering of a victim by a priest to God alone, in testimony of his being the Sovereign Lord of all things. - Cf. Penny Catechism, 275.

It is good to note here that sacrifice is as old as religion itself, having originated simultaneously with religion, this conclusion following from the law of causality because sacrifice is a regular concomitant of every religion. [Cf. Sacrifice | New Advent].

Therefore sacrifice [= offering to a god] is something that comes with and is done in religion.

Proceeding to answer

Does the Judeo-Christian God require sacrifice?

In the Old Testament, as described in the books of the Old Law, God himself laid out how sacrifices were to be offered to him and instituted them as a way of honoring him by offering him some of the creatures that are precious to human beings, in acknowledgment of God's sovereignty and human dependence on him as the Creator. [Cf. Dictionary: SACRIFICE, OLD TESTAMENT | Catholic Culture]. Those sacrifices were either "bloody" or "unbloody" and unbloody sacrifices were never offered for sin or guilt, except at the cleansing of a leper.

From this context it is easier to understand "[i]ndeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins."

Therefore from the laws he established, the Judeo-Christian God requires a bloody sacrifice for the atonement of sin.

Did Christ have to shed his blood in the manner he accepted?

In scenes before this clip in "Pirates of the Caribbean 1: The Curse of the Black Pearl", perhaps one can be led to believe, as I was, that blood required for the curse to be lifted meant someone had to die but all one of the those who could lift the curse needed to do was to spill just some of their blood. Pope Clement VI in explaining the words of St. Paul and St. Peter regarding the oblation of Christ, says: "The innocent Christ, who was immolated on the altar of the cross, shed not a little drop of blood, though this would have sufficed for the redemption of the entire human race, because of the union with the Word but streams of it, like unto a river, so that "from the sole of the foot unto the top of the head, there is no soundness in Him. [Cf. CHRIST THE SAVIOUR — A Commentary on the Third Part of St Thomas' Theological Summa by Reginald Garrigou-Lagrange, O. P.]. A profound idea that St. Thomas Aquinas also sings about in the hymn Adoro Te Devote.

[...] a single drop of which is given all the world from all its sin to save.

Why Christ and in this manner i.e. with streams of his blood to the last drop?

Scripture and Catholic theology explains that since we have to offer the best to God, what possibly could man offer to God that was pleasing to God after the fall? Man was in sin and an infinite God was offended. God provided the solution: God-made-man would be the Expiatory Sacrifice. [Cf. Leviticus and the True Sacrifice by Peter A. Kwasniewski and Gen 22:8 (RSVCE) as type and fore hint].

By his laying down his life for us in the manner, which he does in loving obedience to his Father [cf. Heb 10:1-18 (RSVCE)], therefore "undoing" the disobediece of men and angels, Christ manifested God's love for us [cf. John 3:16 (RSVCE)] and Christ himself showed no one had greater love for us than he [cf. John 15:12-14 (RSVCE)]. It also shows how really bad sin is.

I have a problem with Christ Sacrifice being "needed"

To suggest such a thing suggests that God the father sent his son to die. This was not his will, nor from the begining of creation was this his will. Because of the Nature of God being outside of Time, the whole of History from start to finish is laid out for him in it's immediacy.

To God, all moments of time are present in their immediacy. When therefore he establishes his eternal plan of “predestination,” he includes in it each person’s free response to his grace: “In this city, in fact, both Herod and Pontius Pilate, with the Gentiles and the peoples of Israel, gathered together against your holy servant Jesus, whom you anointed, to do whatever your hand and your plan had predestined to take place.” For the sake of accomplishing his plan of salvation, God permitted the acts that flowed from their blindness. CCC 600

Catholic Church. (2000). Catechism of the Catholic Church (2nd Ed., p. 155). Washington, DC: United States Catholic Conference.

This understanding of God being outside of time does not change his intention for his creation, that being to share in his divinity by pouring himself out in an act of total self gift.

Unable to understand this gift, which was freely given in the Garden, Man at the Fall, rejected the gift of devine sonship, falling and losing the divine Nature, the likeness of God.

Thomas Aquanis addresses this topic in a few articles. Here is one explaining, not easily, how this sacrifice in a way necessary.

FIRST ARTICLE

WHETHER IT WAS NECESSARY FOR CHRIST TO SUFFER FOR THE DELIVERANCE OF THE HUMAN RACE?

Objection 1. It seems that it was not necessary for Christ to suffer for the deliverance of the human race. For the human race could not be delivered except by God, according to Isaias. 45:21: Have not I, the Lord, and there is no God else besides Me? A just God and a Saviour, there is none besides Me. But no necessity can compel God, for this would be repugnant to His omnipotence. Therefore it was not necessary for Christ to suffer.

Obj. 2. Further, what is necessary is opposed to what is voluntary. But Christ suffered of His own will; for it is written (Isa. 53:7): He was offered because it was His own will. Therefore it was not necessary for Him to suffer.

Obj. 3. Further, as is written (Ps. 24:10): All the ways of the Lord are mercy and truth. But it does not seem necessary that He should suffer on the part of the Divine mercy, which, as it bestows gifts freely, so it appears to condone debts without satisfaction: nor, again, on the part of Divine justice, according to which man had deserved everlasting condemnation. Therefore it does not seem necessary that Christ should have suffered for men’s deliverance.

Obj. 4. Further, the angelic nature is more excellent than the human, as appears from Dionysius (Div. Nom. iv.). But Christ did not suffer to repair the angelic nature which had sinned. Therefore, apparently, neither was it necessary for Him to suffer for the salvation of the human race. On the contrary, It is written (John 3:14): As Moses lifted up the serpent in the desert, so must the Son of man be lifted up, that whosoever believeth in Him may not perish, but may have life everlasting.

I answer that, As the Philosopher teaches (Metaph. v.), there are several acceptations of the word necessary. In one way it means anything which of its nature cannot be otherwise; and in this way it is evident that it was not necessary either on the part of God or on the part of man for Christ to suffer. In another sense a thing may be necessary from some cause quite apart from itself; and should this be either an efficient or a moving cause, then it brings about the necessity of compulsion; as, for instance, when a man cannot get away owing to the violence of someone else holding him. But if the external factor which induces necessity be an end, then it will be said to be necessary from presupposing such end—namely, when some particular end cannot exist at all, or not conveniently, except such end be presupposed. It was not necessary, then, for Christ to suffer from necessity of compulsion, either on God’s part, Who ruled that Christ should suffer, or on Christ’s own part, Who suffered voluntarily. Yet it was necessary from necessity of the end proposed; and this can be accepted in three ways. First of all, on our part, who have been delivered, according to John (loc. cit.): The Son of man must be lifted up, that whosoever believeth in Him may not perish, but may have life everlasting. Secondly, on Christ’s part, Who merited the glory of being exalted, through the lowliness of His Passion: and to this must be referred Luke 24:26: Ought not Christ to have suffered these things, and so to enter into His glory? Thirdly, on God’s part, Whose determination regarding the Passion of Christ, foretold in the Scriptures and prefigured in the observances of the Old Testament, had to be fulfilled. And this is what St. Luke says (22:22): The Son of man indeed goeth, according to that which is determined; and (24:44, 46): These are the words which I spoke to you while I was yet with you, that all things must needs be fulfilled which are written in the law of Moses, and in the prophets, and in the psalms concerning Me: for it is thus written, and thus it behoved Christ to suffer, and to rise again from the dead.

Reply Obj. 1. This argument is based on the necessity of compulsion on God’s part.

Reply Obj. 2. This argument rests on the necessity of compulsion on the part of the man Christ.

Reply Obj. 3. That man should be delivered by Christ’s Passion was in keeping with both His mercy and His justice. With His justice, because by His Passion Christ made satisfaction for the sin of the human race; and so man was set free by Christ’s justice: and with His mercy, for since man of himself could not satisfy for the sin of all human nature, as was said above (Q. I., A. 2), God gave him His Son to satisfy for him, according to Rom. 3:24, 25: Being justified freely by His grace, through the redemption that is in Christ Jesus, Whom God hath proposed to be a propitiation, through faith in His blood. And this came of more copious mercy than if He had forgiven sins without satisfaction. Hence St. Paul says (Ephes. 2:4): God, Who is rich in mercy, for His exceeding charity wherewith He loved us, even when we were dead in sins, hath quickened us together in Christ. Reply Obj. 4. The sin of the angels was irreparable; not so the sin of the first man

Thomas Aquinas. (n.d.). Summa theologica. (Fathers of the English Dominican Province, Trans.). London: Burns Oates & Washbourne.

In addition to this discription Thomas speaks of the efficiency of Christs sacrifice here.

Article 6. Whether Christ's Passion brought about our salvation efficiently?

Objection 1. It would seem that Christ's Passion did not bring about our salvation efficiently. For the efficient cause of our salvation is the greatness of the Divine power, according to Isaiah 59:1: "Behold the hand of the Lord is not shortened that it cannot save." But "Christ was crucified through weakness," as it is written (2 Corinthians 13:4). Therefore, Christ's Passion did not bring about our salvation efficiently.

Objection 2. Further, no corporeal agency acts efficiently except by contact: hence even Christ cleansed the leper by touching him "in order to show that His flesh had saving power," as Chrysostom [Theophylact, Enarr. in Luc.] says. But Christ's Passion could not touch all mankind. Therefore it could not efficiently bring about the salvation of all men.

Objection 3. Further, it does not seem to be consistent for the same agent to operate by way of merit and by way of efficiency, since he who merits awaits the result from someone else. But it was by way of merit that Christ's Passion accomplished our salvation. Therefore it was not by way of efficiency.

On the contrary, It is written (1 Corinthians 1:18) that "the word of the cross to them that are saved . . . is the power of God." But God's power brings about our salvation efficiently. Therefore Christ's Passion on the cross accomplished our salvation efficiently.

I answer that, There is a twofold efficient agency--namely, the principal and the instrumental. Now the principal efficient cause of man's salvation is God. But since Christ's humanity is the "instrument of the Godhead," as stated above (Question 43, Article 2), therefore all Christ's actions and sufferings operate instrumentally in virtue of His Godhead for the salvation of men. Consequently, then, Christ's Passion accomplishes man's salvation efficiently.

Reply to Objection 1. Christ's Passion in relation to His flesh is consistent with the infirmity which He took upon Himself, but in relation to the Godhead it draws infinite might from It, according to 1 Corinthians 1:25: "The weakness of God is stronger than men"; because Christ's weakness, inasmuch as He is God, has a might exceeding all human power.

Reply to Objection 2. Christ's Passion, although corporeal, has yet a spiritual effect from the Godhead united: and therefore it secures its efficacy by spiritual contact--namely, by faith and the sacraments of faith, as the Apostle says (Romans 3:25): "Whom God hath proposed to be a propitiation, through faith in His blood.

"Reply to Objection 3. Christ's Passion, according as it is compared with His Godhead, operates in an efficient manner: but in so far as it is compared with the will of Christ's soul it acts in a meritorious manner: considered as being within Christ's very flesh, it acts by way of satisfaction, inasmuch as we are liberated by it from the debt of punishment; while inasmuch as we are freed from the servitude of guilt, it acts by way of redemption: but in so far as we are reconciled with God it acts by way of sacrifice, as shall be shown farther on (49).

Summery of Points

Keeping in mind the attributes of the Persons and Natures of the God head, contemplating that from which humanity fell and that to which Christians shall rise to, the Church understands through its vigilance, That Christ was sent to us, given to us John 3:16 because he loved us. There are many ways in which he could have saved us, including, simply pronouncing our salvation through himself without the Cross at all. Christ however was a teacher, his will and divine purpose united with that of the father, Revealed to us the most efficient way for him, not the only way, or the Necessary way, but the most efficient way, to complete the redemtion of the world. God would become man, and in his goodness free of any sin while living out the Law to perfection, would be killed by his creation. This act, the most humbling example of self gift that could be given to man, the most perfect example of Love for us that ever was is the manor that God choose as the means of our redemption.

Uniting ourselves to this self Gift, making self gifts of ourselves in and through Christ, not on our own is how through the Sacrements of the Church many are drawn towards him in the Eucharist as he is lifted up.

Christ, by his sacrifice on the Cross showed us the love that we rejected and the means by which we can repair our fallen condition, by the same means, joining our lives to his through sacrifice by giving all back to the Father through the Son.