Matt's comment is correct. Jesus is both the "Father" and the "Son" because He has various of roles, some of which fall into a "Father" category and some of which fall into a "Son" category.
Before diving in, I wish to make clear that Mormon doctrine is that Heavenly Father and Jesus Christ are two separate and distinct personages, as based on, e.g., The First Vision of Joseph Smith, where he saw Heavenly Father and Jesus Christ as two separate beings.
Since the two are separate, it can be confusing when Jesus is referred to as "the Father". Referring to Jesus as "the Father" is not saying He is the same person as Heavenly Father.
In my study of Mormon canon, I have identified two ways in which Jesus is "the Father" (there are likely more, but only two I have a good understanding of):
Jesus Christ is the Perfect Son of Heavenly Father, and is a perfect representative of Him. "He that hath seen me hath seen the Father" (John 14:9), "The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise." (John 5:19). As I understand these scriptures, the Father that Jesus is referring to is His Father, not Himself. But the point of these scriptures is that Jesus the Son of God so fully follows what His Father does that He is doing exactly what Heavenly Father Himself would be doing, and so Jesus Christ is "the Father" is that his words, actions, and perfection are a perfect representation of the words, actions, and perfection of His Father.
Jesus Christ is the Father of salvation to those who follow Him. A few verses after the scripture in Mosiah 15 you quote are two verses that teach that Jesus is the Father to those who receive Him and obtain salvation through Him. (Mosiah 15:11-12):
11 Behold I say unto you, that whosoever has heard the words of the prophets, yea, all the holy prophets who have prophesied concerning the coming of the Lord—I say unto you, that all those who have hearkened unto their words, and believed that the Lord would redeem his people, and have looked forward to that day for a remission of their sins, I say unto you, that these are his seed, or they are the heirs of the kingdom of God.
12 For these are they whose sins he has borne; these are they for whom he has died, to redeem them from their transgressions. And now, are they not his seed?
Jesus' role as "the Son" is due to the fact that He is the Son of God. This is the role He is most often viewed in and I don't think I need to add much to help understand how Jesus is a Son. If you need more clarification, let me know.
Later addition: I have found a third way in which Mormon canon identifies Jesus as "the Father". From Mosiah 3:7 (the future tense is because this was a prophesy made before Jesus' birth):
And he shall be called Jesus Christ, the Son of God, the Father of heaven and earth, the Creator of all things from the beginning; and his mother shall be called Mary.
Mormons believe that, although Heavenly Father was the one who planned the creation of heaven and earth, it was the pre-mortal Jesus Christ who actually created heaven and earth. Thus, in this sense Jesus is "the Father of heaven and earth, the Creator of all things from the beginning."