What was the significance of anointing Aaron as high priest?

Answer:

Anointing was a representation of setting something or someone apart for a special purpose. Thus, the altar was anointed before offerings were placed on them to God. "And you shall offer a bull every day as a sin offering for atonement. You shall cleanse the altar when you make atonement for it, and you shall anoint it to sanctify it" (Exodus 29:36). The rest of the tabernacle and the priests were treated the same way. "And you shall take the anointing oil, and anoint the tabernacle and all that is in it; and you shall hallow it and all its utensils, and it shall be holy. You shall anoint the altar of the burnt offering and all its utensils, and consecrate the altar. The altar shall be most holy. And you shall anoint the laver and its base, and consecrate it. Then you shall bring Aaron and his sons to the door of the tabernacle of meeting and wash them with water. You shall put the holy garments on Aaron, and anoint him and consecrate him, that he may minister to Me as priest" (Exodus 40:9-13).

Thus the anointing showed that Aaron was set apart (sanctified) for a special or holy purpose in God's service.