It seems at first a little odd that the writer of Hebrews tells his readers to “stir one another to love and good deeds.” All of chapter 10 prior to that is spent talking about sprinkled blood and clean consciences. It speaks of the confidence we can have in Christ our high priest to represent us before the throne of consuming fire.
After such glorious, albeit terrifying, thoughts, the writer turns immediately and without warning to his admonition that we consider how to stir one another up to good works. The link is not immediately apparent.
However, I think the link is sin. We have no idea just how heinous sin is in God’s sight. But it was the awfulness of sin which brought Christ’s high priestly work about, and it is the awfulness of sin which threatens our soul’s well-being. Therefore, it is against the awfulness of sin that the writer of Hebrews warns us when he says we should consider how to stir one another up to love and good works. We must fight sin with love and good works, and we must not try to fight that battle alone. We need one another.