We are a kingdom of priests. We have been washed in our baptism, clothed with the righteousness of Christ, and anointed with the Spirit. The OT priests also were washed, clothed, and anointed. Then they had the sacrificial blood placed on them to set them apart as priests. This goes deeper in the New Covenant, where living water flows from within us. We are clothed with our own acts of righteousness done by the Spirit. And we receive the blood of Christ into ourselves, united with His death, dying in His death to our sin. So we sit as priests, baptized, clothed and anointed by God, a new people.
All this can come to us because Jesus gave it up in His passion and death. He cleans us with water, but was thirsty and dry Himself on the cross. He clothes us, but was naked Himself on the cross. He gives us His life-giving blood, letting Himself bleed on the cross. The cross was the fulfillment of all righteousness, the fulfillment of His baptism.