The robe, ring & sandals help show the father’s high level of love, honor and authority for the son. The robe and the ring are symbolic of how well the father will be treating his son (i.e. somewhat like Jacob and Pharaoh treated the favorite son Joseph).
Jacob honored Joseph by getting him a long tunic, and the jealous brothers saw how Jacob was the favorite son. (Gen 37:3-4)
They hated him so much they sold him into Egypt.
When in Egypt, Joseph heard of Pharaoh’s dreams and interpreted them for him, saying there would be seven years of abundance followed by seven years of famine. With that Pharaoh put Joseph in charge of Egypt, and while doing so, Pharaoh placed his own ring on Joseph’s finger. He then had him dressed in robes of fine linen. (Gen 41:41-43)
“In that ancient world a ring was not simply a matter of vanity. It was no mere item of excess wealth. A ring carried as great a significance to them as an engagement or wedding ring does in our world.” (prodigalprof)
“What does this giving of a ring mean? It is the granting of authority to a person. Whoever has such a ring has the power of attorney for his master. He has authority, his master’s authority, to make decisions and to help the master govern his realm. And when the father places the ring on the hand of his son, he not only welcomes him back home as a son, as was indicated by the robe, but he welcomes him back to responsibility and authority.”
The sandals signified he was NOT a servant but the son of the father. Servants wore no sandals so when the father refused to let his son be a mere servant, this is indicative of the benevolence and love of the father.