Short answers, pulled from the story.
The threefold office of Christ refers to his roles as prophet, priest, and king. Ancient leaders received authority through anointing rituals that sanctioned appointments to these positions. The Hebrew word messiah literally means anointed one connecting directly to these ceremonies.
Jesus claimed his words belonged to the Father who sent him rather than using the messenger formula Thus says the Lord. He served as the source of revelation instead of just a messenger bringing messages to people. John 14:24 records Jesus saying These words you hear are not my own.
Christ offered himself as sacrifice with indestructible power overcoming human weakness. He made one sin offering unlike Old Testament priests who continually offered sacrifices. His atoning death stands at the heart of his priestly work allowing humanity to have a living relationship with God.
John Calvin initially treated Christ's saving work under two heads of king and priest before presenting all three offices later. He did not fully present the three offices until the third edition of Institutes published in 1559. Eusebius worked out this threefold classification earlier but Calvin developed it further.
On the 11th of April 1844 the Council of Fifty voted to sustain Smith as Prophet Priest and King. This unique application contrasts sharply with Roman Catholic definitions found in official catechisms. Smith's leadership claimed all three offices simultaneously unlike traditional Christian doctrine.