How does the Israelites know that Moses is a prophet of God?

There are three points:

1. The message of the promised land of Canaan;
   The message of Moses is consistent with that of the promises of God to Abraham, Isaac and Jacob. Moses confirmed (Exodus 6:8) God's promise to Abraham (Genesis 17:8), Isaac (Genesis 26:3) and Israel (Genesis 35:12) to give them the land of Canaan by leading them out of Egypt to Canaan.
   And I will give unto thee, and to thy seed after thee, the land wherein thou art a stranger, all the land of Canaan, for an everlasting possession; and I will be their God. (Genesis 17:8 KJV)
   Sojourn in this land, and I will be with thee, and will bless thee; for unto thee, and unto thy seed, I will give all these countries, and I will perform the oath which I swore unto Abraham thy father; (Genesis 26:3 KJV)
   And God said unto him, Thy name is Jacob: thy name shall not be called any more Jacob, but Israel shall be thy name: and he called his name Israel. And God said unto him, I am God Almighty: be fruitful and multiply; a nation and a company of nations shall be of thee, and kings shall come out of thy loins; And the land which I gave Abraham and Isaac, to thee I will give it, and to thy seed after thee will I give the land. (Genesis 35:10-12 KJV)
   And I will bring you in unto the land, concerning the which I did swear to give it to Abraham, to Isaac, and to Jacob; and I will give it you for a heritage: I am the LORD. (Exodus 6:8 KJV)


2. The visible Shekinah Glory of God;
   The visible Shekinah Glory of God lead the Israelites by day in a pillar of a cloud and by night in a pillar of fire. The Glory of the LORD came down and covered the Mount Sinai for six days and on the seventh day called Moses to receive the Ten Commandments.
   And the LORD went before them by day in a pillar of a cloud, to lead them the way; and by night in a pillar of fire, to give them light; to go by day and night: He took not away the pillar of the cloud by day, nor the pillar of fire by night, from before the people. (Exodus 13:21-22 KJV)
   And the glory of the LORD abode upon mount Sinai, and the cloud covered it six days: and the seventh day he called unto Moses out of the midst of the cloud. (Exodus 24:16 KJV)


3. The miracles performed in the sight of all Israel.
   The miracles performed in the sight of all Israel in the deliverance from Egypt and in the Wilderness.
   And there arose not a prophet since in Israel like unto Moses, whom the LORD knew face to face, in all the signs and the wonders, which the LORD sent him to do in the land of Egypt to Pharaoh, and to all his servants, and to all his land, and in all that mighty hand, and in all the great terror which Moses showed in the sight of all Israel. (Deuteronomy 34:10-12 KJV)