In what sense is Christ the mediator of the covenant? He is not like Moses, who stood between God and the people of Israel, "to show" them "the word of the Lord" (Deut. 5:5), to receive the law, the living oracles, and deliver them to the people. Christ indeed is the revealer and proclaimer of his Father’s mind and will and the dispenser of the covenant of grace.
Christ is a mediator of reconciliation; such as one who goes between two parties with a disagreement to bring them together and in some way or other reconcile them to each other. God is one party, the offended party, and man is the other, the offending party. Christ is the mediator between them both to bring them together, who are through sin at as great distance as earth and heaven. He is like Jacob’s ladder that reaches both and joins them together and makes peace between them.
This work he performs not merely by way of entreaty, as one man may entreat another to lay aside his resentment against an offender or as Moses entreated God to forgive the Israelites. However commendable this may be for one man to intercede with another, or with God for an offender in such a manner, it seem too low and mean an office for Christ, the Son of God, merely to entreat his Father to lay aside the marks of his displeasure against a sinner.
Therefore, Christ acts the part of a mediator by proposing to his Father to make satisfaction for the offense committed, and so appease injured justice. Christ is a mediator of reconciliation in a way of satisfaction. Reconciliation in this way is Christ’s great work as mediator. This is what was proposed as a covenant and what he agreed to do, and therefore he is called the mediator of the covenant.
Adapted from A Body of Doctrinal Divinity, Book 2, Chapter 11, by John Gill.