The symbolism looks forward to Jesus' ultimate cleansing of those who believe. The bird put to death represents Christ's death on the cross for our sins, and the bird dipped in that blood and released represents His resurrection and ascension into heaven where He continues to intercede on our behalf.
When we are saved, Jesus declares us to be free of sin, just as the priest declared the leper free of leprosy. The elaborate purification process may represent our on-going sanctification. Though our salvation is secured by the finished work of Christ, in gratitude and understanding of our nature, we continue to identify and root out every vestige of sin that remains through the conviction of the Holy Spirit.