Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us. For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God? ~ Hebrews 9:12-14

The audience to which the writer of Hebrews is making his appeal were his fellow Hebrew believers who had come out of the old temple worship according to the law.

These Hebraic believers were being tested in their faith and the temptation to abandon Christ for the old worship was a real thing. This is one of the reasons there is such an emphasis on the superiority of the sacrifice of Jesus in contrast to the animal sacrifices under the law. A return to animal sacrifices after coming to the knowledge of Christ would be considered apostate.

The temple worship with its carnal ordinances, which included the animal sacrifices, is specifically what the writer of Hebrews is referring to when he speaks of the dead works, from which they had been cleansed by the blood of Christ. ~ Hebrews 9:1-14

Of course, for us today, dead works could refer to other things such as religious duties, but for the audience at the time of the writing of Hebrews, the dead works was the old worship (the carnal ordinances) which had been prescribed under the law with its priesthood and sacrifices.