There are three key positions in ancient Israel, that of Prophet, Priest, and King. A prophet would hear from God and speak to the people. A priest would hear from the people and intercede for them to God. A king would, of course, rule.
Jesus fulfills all of these positions. He spoke of His own death and even of future events, like when He spoke of the destruction of Jerusalem (see Mark 13:2) and of end times.
He was most certainly a priest, though not of the Levitical (and especially not the Aaronic) priesthood, but the Melchizedek priesthood. (see, e.g., Hebrews 7)
He also is the King of Kings (Revelation 19:16), and heir to David's throne (2 Samuel 7:12-16, Mark 10:47, et. al.)
So, yes, this is speaking of Jesus.
To be more precise about how exactly Jesus was a prophet like Moses, note the following similarities from this article:
1 Both Moses and Jesus were born when Israel was under bondage (Moses in Egypt Ex.1:8,10, and Jesus during Israel's Roman bondage. Mt.2:1,Lk.2:1-2)
2 Edicts were issued by the gentile powers of both Pharaoh and Herod for their death at birth. (Ex.2:3, Mt.2:15)
3 Both were miraculously delivered from death during their infancy from their enemies of Israel (Ex.2:3, Lk.2:7;Mt 2:14,15) and were preserved in childhood.
4 Both had been born and found in unusual places. Moses was found in the water floating in a basket of bulrushes (Ex.2:3) Jesus in a manger (cave) wrapped in burial cloths (Lk.2:7)
5 Both Moses and Jesus were protected and preserved by the faith of their parent. Moses was hidden by faith (Ex.2:2-3 and Heb.11:23) Jesus’ life was preserved by his parents obeying the message in a dream. (Mt.2:13-14)
6 Both Moses and Jesus stayed safely in Egypt for a time. Moses (Ex.2:10) Jesus dwelt safely in Egypt (Mt.2:14-15)
7 Both had their brethren and family speak against them. Moses had Mariam and Aaron speak against him taking a Cushite wife (Numb.12:1) Jesus had his family criticize and reject him (Mt.13:54-57, Jn.7:3)
8 Both Moses and Jesus were reared in the house that was not their natural parents. (Moses in the house of Pharaoh (Ex.2:10) Jesus by Mary the mother of his humanity and his stepfather Joseph.
9 Both predicted Israel's history (Deut.28:15-28; Mt.23:34,24:1,2,8,34)
10 Both spoke out against Israel's enemies and persecutors. (Deut.23:3-4; Mt.25:41-48)
11 Both were considered a snare to Israel (Ex10:7; 1Pt.2:8=Isa.8:14)
12 Both cleansed leprosy. From Moses’ time no one that was Jewish was cleansed until Jesus came (Numb.12:10-16, Mk.1:40-41). This is a very significant factor that proves Jesus was the prophet Moses spoke of, since no other cured one of the brethren of this disease from Moses' time.
13 Both were used of God to feed Israel miraculously, Moses with manna in the wilderness (Ex.16:14-17) Jesus with the five loaves and two fish fed four and five thousand two times. (Mt.14:19-21) Jesus also called himself the true manna that came from heaven in Jn.6, comparing himself to the miracle that sustained Israel alive through the desert.
14 Both had the forces of nature obey them (the seas) Moses (Ex.14:21-22) Jesus (Mt.8:26-27)
15 Each had seventy helpers Moses (Num.11:16-17) Jesus ( Lk.10:1).
16 Both fasted 40 days and 40 nights in the wilderness to bring a covenant to Israel. Moses on the top of Mt. Sinai (Ex.24:18, 34:28, Deut.9:9) Jesus in the desert as the Son of God (Mt.4:2)
17 Both had a face to face relationship with God unlike any other person. Moses (Ex.33:9-11, Deut.34:10; Numb.12:7-8) Jesus was with God=face to face from eternity (Jn.1:1,18)
18 God spoke audibly and directly from heaven to both Moses and Jesus. Moses (Ex.20:22, 24:12-16) Jesus (Mt.3:17; Jn.12:28)
19 Both were God's spokesman to and for the people. Moses (Ex.9:35; Numb.12:2) Jesus (Heb.1:1-3). They Spoke as oracles of God (Moses Deut. 18:18) Jesus (Jn.14:24, 5:24)
20 Both Moses and Jesus reflected and shown the glory of God. Moses reflected his glory temporarily. (Ex.34:29-35; 2 Cor.3:7-14) Jesus at the transfiguration, his face shown brighter than the sun revealing his true nature (Mt.17:2; Jn.1:14)
21 Both were known for their humility and meekness as God's servants. Moses (Numb.12:3) Jesus (Mt.11:29; Phil.2:3-8)
22 Both contended with masters of deception and darkness Moses with Pharaohs magicians (Ex 7:11, 1 Tim.3:8) Jesus with the Devil (Mt 4:1).
23 Both prayed for the people intercessory prayers and were willing to bear the consequences of the people's sins. Moses asked to be blotted out of the book of life for the peoples sake (Ex.32:32-33). Jesus asked for them to be kept from falling away (Jn. 17:9-17). Jesus asked for those to be forgiven while he bore the consequences in their stead. (Lk.23:34, 2 Cor.5:19, 1 Pt.2:21-24, Isa.53:8)
24 Both Moses and Jesus were rejected by their own brethren for a time and accepted by the gentiles. Moses (Ex.2:14-22, 32:1) Jesus was rejected by his own people and received by another who were not his own. (Isa.53:3; Mt.12:21; Mk.6:4; Lk.20:9-17, Rom.11:20)
25 Both Moses and Jesus established a priesthood. Moses began the Aaronic priesthood which was temporary under the law (Lev.9; Numb.8:20-26; Heb.9:19-22). Jesus established a eternal priesthood under the new covenant of grace which He alone functions as the high priest forever. (Heb.7:17,19,23,25-28; 9:12)
26 Both sprinkled the blood of the covenant on the altar and the people. Moses (Ex.24:7-8; Lev.8:19) Jesus (Heb.9,12:24; 1Pt.1:2)
27 Both were sent by God to reveal His name, person and law to the people. To Moses God said to tell them I Am sent you. (Ex.3:13-14) Jesus said God sent him as his exact representative revealing his name (I Am) and nature to the people. (Jn.8:42; 17:6,11-12; Col.2:9; Heb.1:3)
28 Both were involved in giving the covenant to the people Jn.1:17 the law came through Moses but grace and truth came through Jesus Christ.” At the feast of Pentecost Moses gave the law. At the feast (Pentecost) in the new covenant Jesus gave the Holy Spirit (Acts 2).
29 Both brought deliverance to the Jewish people. Those who followed Moses out of the bondage of slavery to Egypt. (Ex.3:7-8,10;12:31-33,42) Jesus brought people out of a greater bondage, slavery to sin. (Rom.3:24-25,6:6-7,8:2-4; Eph.1:7;Heb.9:26)
30 Both had an angel guard their graves. After Moses died Michael the Arch-angel guarded his body. (Jude 9) And when Jesus rose an angel guarded his tomb. (Mt.26:2-6)
31 Both re-appeared after they died, Moses ( Mt 17:3 with Jesus before his death) Jesus (Acts 1:3)