Hebrews 10:15-18 meaning

The Pauline Author returns to Jeremiah 31 here. In chapter 8, he quoted the chapter broadly to show the Old Testament prophecies that pointed to the New Covenant. Here in Hebrews 10:15-18, the Pauline Author focuses on Jeremiah 31:33, highlighting the complete forgiveness found in Christ: 

And the Holy Spirit also testifies to us; for after saying, "This is the covenant that I will make with them after those days, says the Lord: I will put My laws upon their heart, and on their mind I will write them." He then says, "And their sins and their lawless deeds I will remember no more." (vv 15-17)

In the verses prior to these, he has established that Christ is a better high priest than earthly priests and that His sacrifice was once for all, while earthly sacrifices are never-ending. They must be repeated year after year. In light of that, the Pauline Author quotes Jeremiah, attributing the prophecy there to the Holy Spirit, the third person of the Trinity, testifying to us (humanity) about this New Covenant. 

And, the Pauline Author points out, this New Covenant is one where God puts His law in His people's minds and hearts, and where He will forget about all their sins and their lawless deeds. In this New Covenant, the law is no longer a list of rules to obey, rather, the Holy Spirit lives in every believer, guiding our minds and hearts. God does not remember our past sins because Jesus's perfect sacrifice was the perfect atonement for all wrongdoing. This doesn't mean He loses recollection. It means He chooses to no longer take it into account.

Therefore, the Pauline Author concludes that where there is forgiveness of these things, there is no longer any offering for sin (v 18). Since there is total forgiveness through Christ, there is no reason to make animal offerings to take away sin under the New Covenant. The continuous offerings do not take away sin. This is a key point the Pauline Author keeps hammering, because apparently the Jewish believers receiving this letter have begun depending on animal sacrifice to take away sins they are willfully committing.

Christ's offering accomplished everything. The Pauline Author stresses that we are in this New Covenant, and God has put His law inside us. God is guiding us through His Spirit, and we are forgiven of all prior sins. The Old Covenant, the law, the priests, and animal sacrifices served their purpose—and have been displaced. This demonstrates the folly of beginning to rely and focus upon religious ceremony. It is fine to do it, but it is not fine to rely upon it. The obedience of faith comes from the heart, and from a clear conscience.

Christ is the greater High Priest and His offering was the greatest sacrifice that could be made because now, for those who believe in Him, their sins are forgiven and God's law is put into their hearts. It is the greatest of folly to turn from God in our hearts while looking to religious practice to cover sin. It is only Jesus who justifies us in the sight of God.