What does Hebrews 10:15 mean?

Since Hebrews was written to persecuted Jewish Christians, the writer often makes use of quotations from the Old Testament. Use of these Jewish Scriptures is meant to reassure the reader that these are not new ideas. On the contrary, the ministry of Jesus Christ is exactly what God was predicting and promising in the Old Testament. The old covenant was a symbol of the new covenant, meant to teach and prepare us. The new covenant was promised, directly, by God. This is supported by references to figures such as Melchizedek (Hebrews 7:11–17) and prophets such as Jeremiah (Hebrews 8:7–13).

This is why this verse points out that it is the Holy Spirit who speaks to us. The segment quoted in the next verse is from Jeremiah 31:33, first quoted in Hebrews chapter 8. In the Jeremiah passage, God predicted a new covenant which would be written on the hearts and minds of the people. This new covenant, in contrast to the old, would mean a real resolution for human sin.