As if the people required any more evidence of the divinity of the Savior, he places himself in the context of fulfilled revelation by citing the prophecy given to Moses. The citation is obviously not given to prove something the people did not already believe, but rather it showed the deep respect that the Savior had for the scriptures by making certain to highlight his place in the fulfillment of them.
Reference: The ultimate reference of this verse is in Deuteronomy:
Deuteronomy 18:15-16
15 ¶ The LORD thy God will raise up unto thee a Prophet from the midst of thee, of thy brethren, like unto me; unto him ye shall hearken;
However, the immediate reference to this comes from Acts where we find the more specific declaration following the reference:
Acts 3:22-24
22 For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you.
23 And it shall come to pass, that every soul, which will not hear that prophet, shall be destroyed from among the people.
24 Yea, and all the prophets from Samuel and those that follow after, as many as have spoken, have likewise foretold of these days.
The reference here is unquestionably influenced by the New Testament. The meaning is certainly understood in the Book of Mormon, and indeed, the opening scenes of the Book of Mormon have Lehi declaring:
1 Nephi 10:2-5
2 For behold, it came to pass after my father had made an end of speaking the words of his dream, and also of exhorting them to all diligence, he spake unto them concerning the Jews—
3 That after they should be destroyed, even that great city Jerusalem, and many be carried away captive into Babylon, according to the own due time of the Lord, they should return again, yea, even be brought back out of captivity; and after they should be brought back out of captivity they should possess again the land of their inheritance.
4 Yea, even six hundred years from the time that my father left Jerusalem, a prophet would the Lord God raise up among the Jews—even a Messiah, or, in other words, a Savior of the world.
5 And he also spake concerning the prophets, how great a number had testified of these things, concerning this Messiah, of whom he had spoken, or this Redeemer of the world.
The foundation of Nephite understanding is that there were a “great number’ of prophets who had testified of the Messiah. Nevertheless, the particular citation here is to the New Testament, and clearly an artifact of the translation rather than the plate text, even though the meaning is quite at home on the plate text.