After the descriptions of the priestly vestments, the Torah discusses the inauguration ceremony for the priests. Aaron and his sons are instructed to place their hands on various sacrifices and to be anointed with special oil. Moses is also instructed to  

Slaughter the ram, and take some of its blood and put it on the lobe of Aaron’s right ear and on the ridges of his sons’ right ears, and on the thumbs of their right hands, and on the big toes of their right feet; and dash the rest of the blood against every side of the altar roundabout” (verse 20). 

Why is the blood sprinkled on these specific extremities? 

One possibility is that the ritual links the altar with the priests. Moses is also commanded to sprinkle blood on the horns of the altar and the sides. The priests, who are primarily responsible for the sacrificial rites also have blood sprinkled on their extremities. 

Another answer is that the three sprinklings match three different commandments to the priests. One that they listen to God (ear), two is

that they perform the proper sacrificial rites (thumb) and finally that they walk in the way of God (toe). Yet another possibility is to emphasize that the priests are still human beings despite their involvement in the sacrificial rites, which are bloody and almost animalistic. Humans are unique with their cognitive abilities (ear), they have opposable thumbs (thumb) and they are bi-peds (big toe).  

The three body parts also represent three sections of the body- the head, the trunk, and the legs. The priests then are being purified throughout their bodies and are reminded that their entire body now belongs to the service in the mishkan. 

Josh has taught many classes on Tanach throughout the years and currently in the New Rochelle 929 group, and is an in-house attorney in New Jersey.