“Those who have had a bath need only to wash their feet; their whole body is clean." John 13.10
To understand the significance of Jesus' priesthood, one needs to grasp the origin and nature of sin. The Bible describes how God made the world and humankind, represented by Adam and Eve, as its crowning glory: creatures made in his own image, with the gift of freewill, without which it is impossible to love because love cannot be forced. It was always possible, though not inevitable, that Adan and Eve would eventually fall out of love with God and go their own way. Yet, on account of the essential goodness of creation, if it was to appear at all, such rebelliousness would have to come from an external source. According to the Book of Genesis, the culprit was Satan. Where he came from is another story! Suffice to say that his influence over Adam and Eve resulted in their decision to disobey God, thereby ruining the harmony of creation.
To contain the damage God expelled Adam and Eve from his presence and introduced mortality to life, so that suffering and evil would be finite. For humankind, the most tragic consequence was our spiritual deafness and blindness. With the fall we lost the instinctive awareness of God's presence. Even when, in his grace, God reached out through the prophets, there was still a chasm between humankind and God, before whose purity we withered in our sinfulness. Herein lie the origins of the temple system of animal sacrifices, offered by a priestly caste who were set aside for their special role in approaching God on behalf of his fallen people. Of course it was never enough because human priests share the rest of creation's fallenness, as do even the least blemished sacrificial birds and animals.
How to bridge the gap and reconcile creation to our Maker? Send a perfect priest to offer the perfect sacrifice...