Note the association of the word "prophet" with the phrase, "I will put my words in his mouth." This is what God told Moses He would do, so a chain of communication is set up—from God to Moses, from Moses to Aaron, and from Aaron to Pharaoh or to the people.
Contrary to what it shows in The Ten Commandments movie, the Bible suggests that Aaron did the bulk of the speaking before the people rather than Moses. This does not mean that Moses was excluded from speaking to the people, because eventually, even though it is likely that he never overcame his lack of eloquence (Exodus 4:10), he nonetheless became secure in his position as the leader. As the forty-year trial went on, he more often spoke directly to the people. When Israel finally got away from Pharaoh, Moses probably did the bulk of the speaking before the people, and Aaron faded into the background in that regard.
Every other prophet, except Christ, only built on the foundation laid in Moses. These verses particularly foretell of Christ, but it applies in principle to all the prophets that followed Moses. They all were spoken to by God, and they in turn did what Moses did: delivered the message to the ones it was addressed to.
Until New Testament times, prophets have been God's way of reaching the people. Whenever the people needed a prophet or a mediator with God, as He says in verses 16-17, God would raise up a prophet and put His words in his mouth.