The phrase "blood of Christ" refers to the Work of Jesus Christ on the Cross. Blood represents judgment -- the judgment for our sins while Christ was bearing them on the Cross (His spiritual death).
The physical death of the animal in the Old Testament sacrifices is analogous to the spiritual death of Christ.
"Who his own self bare our sins in his own body on the tree, that we, being dead to sins, should live unto righteousness: by whose stripes ye were healed." 1 Pet. 2:24
"For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him." 2 Cor. 5:21
Every animal sacrifice spoke of the Work of Christ.
"The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world." John 1:29
"He was oppressed and he was afflicted, yet he opened not his mouth: he is brought as a lamb to the slaughter, and as a sheep before her shearers is dumb, so he openeth not his mouth." Isa. 53:7
The animal had done nothing wrong but was judged anyway. In the judgment, the animal died physically. Jesus Christ had done nothing wrong, in fact He was impeccable. In His judgment, the sins of the whole world were poured out on him. The phrase "blood of Christ" sets up a representative analogy with the animal sacrifices of the Old Testament. The Workx of Christ was presented by means of these ceremonial practices.
[ For more information, study the topics of Reconciliation, Propitiation, The Barrier, and Furniture of the Tabernacle. ]
After Jesus Christ's work on the Cross was completed, He said, "It is finished!" Then He said, "Father, into thy hands I commend my spirit." He dismissed His spirit; He did not bleed to death on the Cross.
"Therefore doth my Father love me, because I lay down my life, that I might take it again. No man taketh it from me, but I lay it down of myself. I have power to lay it down, and I have power to take it again..." John 10:17,18.
"And when Jesus had cried with a loud voice, he said, Father, into thy hands I commend my spirit: having said this, he gave up the ghost." Luke 23:46
After Jesus Christ had died physically, the soldier ran a spear into His side, and blood and water (serum) came out. The separation of the serum and blood indicated that He had already died physically. Hence, he did not bleed to death. And His throat was not cut as was the animal's in the sacrifice. READ John 19:31-34
So, the bleeding of the animal is an analogy to the spiritual death of Jesus Christ (His judgment for us). The judgment of the animal was physical death. The judgment of Christ was spiritual death.
"And having made peace through the blood of his cross, by him to reconcile all things unto himself; by him, I say, whether they be things in earth, or things in heaven." Col. 1:20
"And their sins and iniquities will I remember no more. Now where remission of these is, there is no more offering for sin. Having therefore, brethren, boldness to enter into the holiest by the blood of Jesus." Heb. 10:18-20
This judgment served to remove the Barrier between man and God. There is a meeting between the two sides of the Cross, and that meeting is in the phrase "blood of Christ".
The teaching of the death of Christ for the sins of the world was, to Old Testament people, by the use of the blood sacrifices. And the significance of the analogy is that it relates us directly to the millions of people who were saved before the time of Christ. They were saved just as we are, by grace provision and faith in the Lord Jesus Christ.
There is no longer a need for animal sacrifices, because Christ's sacrifice on the Cross is once for all.
"And almost all things are by the law purged with blood; and without shedding of blood is no remission." Heb. 9:22
The blood of Christ is the basis for all of the salvation doctrines of the New Testament, including the following:
Redemption
"Forasmuch as ye know that ye were not redeemed with corruptible things, as silver and gold, from your vain conversation received by tradition from your fathers, but with the precious blood of Christ, as of a lamb without blemish and without spot:" 1 Pet. 1:18,19
"In whom we have redemption through his blood, the forgiveness of sins according to the riches of his grace." Eph. 1:7
Justification
"Much more then, being now justified by his blood, we shall be saved from wrath through him." Rom. 5:9
Imputation -- God's righteousness is imputed (credited) to us because our sin is imputed (credited) to Christ. 2 Cor. 5:21 (above). God looks at the believer and sees a righteousness equivalent to His own. This is possible because our sins were judged at the Cross and we were given righteousness by Grace.
Propitiation
"Whom God hath set forth to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God:" Rom. 3:25
Cleansing and Restoration to Fellowship
"But if we walk in the light, as he is in the light, we have fellowship with one another, and the blood of Jesus Christ cleanseth us from all sin. If we say that we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sin, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness." 1 John 1:7-9
[ Read Leviticus chapters 4 and 5 for an account of the two offerings related to restoration to fellowship. ]