Before Aaron and his sons can serve as priests, three things are required: There must be a washing, a covering, and an anointing. The washing is not some perfunctory sprinkling of water, but a bathing, and this was something that had to be done to them. Having been cleaned, they could put on the sacred garments, and then receive an anointing and consecration into the office of the priesthood.
Note that nothing, no matter how elaborate or finely prepared, was good enough – in and of itself – to be used in God’s worship. Everything had to be “cleansed” and marked as holy for that special use and nothing, marked for that special service to God, could be used in a worldly fashion for the pleasure of mankind. Moving through this section, you see that God likes things clean and smelling good – perhaps the genesis of that old phrase (not in the Bible) “cleanliness is next to godliness.”
Having been washed to be set apart for service to the Lord, the priests nevertheless had to keep washing as they approached the Lord. To do so without washing resulted in death.
Two points about this strike me: First, the notion that one can come into the Lord’s presence to serve Him without any thought about it at all or special preparation (“come as you are”) is totally foreign to the theology of the Bible. Even in the New Testament, Paul condemns people who observe the Lord’s Supper without preparing or giving it thought.
Second, how alike Aaron’s consecration into the priesthood is the consecration of every Christian: we are washed in baptism (Acts 22:16), clothed with Christ (Galatians 3:27), and anointed by God’s Spirit (Acts 2:38). Having been consecrated for His service, we are special. Nothing about us should evidence we are anything but the uncommon people God has called us to be.