The mighty change spoken of is the same transformation from carnal to spiritual that occurs when we are born again. Because of denominational inferences, latter-day saints don't refer to themselves as "born again Christians," but the Book of Mormon teaches us that this mighty change must take place in order for us to enter into the kingdom of God (Mosiah 27:26).
In order to explain the doctrine of being born again, we begin with the interview between Nicodemus and the Savior:
'Except a man be born again, he cannot see the kingdom of God.
Nicodemus saith unto him, How can a man be born when he is old? Can he enter the second time into his mother's womb, and be born?
Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. That which is born of the flesh is flesh; and that which is born of the Spirit is spirit' (Jn 3:3-7).
The usual explanation of this doctrine is that one must be baptized and receive the gift of the Holy Ghost by the laying on of hands in order to enter the kingdom of God. This "Sunday School answer" doesn't totally explain the concept because it is possible to receive the gift of the Holy Ghost without being born of the Spirit. The former is an ordinance; the latter is a spiritual transformation. The two can occur at the same time, but they usually don't. Let's take the people of Benjamin for example. These people were presumably already baptized. Yet, they are blessed with a spiritual transformation based upon their faithfulness and the covenant that they made at the time of Benjamin's speech. Therefore, the spiritual rebirth did not transform them from carnal to spiritual at the time of their baptism. This event occurred later, and it is the same for many members today. They are born again not at the time of baptism and confirmation but after they have made the decision to humble themselves before the Lord and promise to do his will.
The next concept which requires elucidation is the timing of being born again. Often the scriptures speak of this happening over a relatively short time period of a day or two. Benjamin's people, Alma, Enos, and king Lamoni are examples. However, the vast majority of members make this mighty change over a longer time period—sometimes over months or years. Bruce R. McConkie said, "(For most of us) this process is usually slow. The unusually quick ones make their way into scripture." (BYU Speeches of the Year, 1976). The process, whether it takes a day, a year, or a decade, is characterized by a spiritual transformation—'changed from their carnal and fallen state, to a state of righteousness, being redeemed of God, becoming his sons and daughters. And thus they become new creatures' (Mosiah 27:25-26). Joseph Smith said, "the effect of the Holy Ghost upon a Gentile, is to purge out the old blood, and make him actually of the seed of Abraham. That man that has none of the blood of Abraham (naturally) must have a new creation by the Holy Ghost." (Teachings, p. 150).
The Spirit changes the very nature of the individual such that they have no more disposition to do evil, but to do good continually. They receive his image in their countenances and feel to sing the song of redeeming love (Alma 5:14, 26). This is attended by the manifestations of the Spirit. The gift of tongues was present on the day of Pentecost. The people of Benjamin declared that they had the spirit of prophecy, saying, were it expedient, we could prophesy of all things (v. 3). Nephi declared that the baptism of fire is followed by the ability to speak with the tongue of angels (2 Nephi 31:13), which is also a representation of the spirit of prophecy.
Finally, we become the sons and daughters of Christ, spiritually begotten by his eternal sacrifice (v. 7). He forever becomes our Father and our God. In this context, one must ask how appropriate it is to refer to Jesus as our elder brother? Certainly, in the pre-mortal existence, He was our elder brother, but if we have been born again, he can be our elder brother no longer. By virtue of the atonement he becomes the Father of our spirits, for behold, this day he hath spiritually begotten you (v. 7).
Bruce R. McConkie
"Mere compliance with the formality of the ordinance of baptism does not mean that a person has been born again. No one can be born again without baptism, but the immersion in water and the laying on of hands to confer the Holy Ghost do not of themselves guarantee that a person has been or will be born again. The new birth takes place only for those who actually enjoy the gift or companionship of the Holy Ghost, only for those who are fully converted, who have given themselves without restraint to the Lord. Thus Alma addressed himself to his 'brethren of the church,' and pointedly asked them if they had 'spiritually been born of God,' received the Lord's image in their countenances, and had the 'mighty change' in their hearts which always attends the birth of the Spirit. (Alma 5:14-31.)" (Mormon Doctrine, p. 101)
Bruce R. McConkie
"When the Holy Ghost falls upon a worthy recipient, it has the effect of pouring out pure intelligence upon him; all is calm and serene; the still small voice speaks peace to the spirit within man; and the sanctifying, cleansing power of the Spirit begins to manifest itself. (Teachings, pp. 149-150.)" (Doctrinal New Testament Commentary, vol. 1, p. 142)
Joseph F. Smith
"What a glorious condition was this! a condition in which the Lord God Omnipotent, by the power of His Spirit, had wrought a mighty change in the hearts of that people, that they no longer had any desire to do evil, but were filled only with a fervent desire to do that which was good. This was indeed a great change, and yet it is precisely that change that comes today to every son and daughter of God who repents of his or her sins, who humble themselves before the Lord, and who seek forgiveness and remission of sin by baptism by immersion, by one having authority to administer this sacred ordinance of the Gospel of Jesus Christ. For it is this new birth that was spoken of by Christ to Nicodemus as absolutely essential that men might see the Kingdom of God, and without which no man could enter into the Kingdom. Each of us can remember, perhaps, the change that came into our hearts when we were baptized for the remission of our sins…The feeling that came upon me was that of pure peace, or love and of light. I felt in my soul that if I had sinned-and surely I was not without sin-that it had been forgiven me; that I was indeed cleansed from sin; my heart was touched and I felt that I would not injure the smallest insect beneath my feet. I felt as though I wanted to do good everywhere to everybody and to everything. I felt a newness of life, a newness of desire to do that which was right. There was not one particle of desire for evil left in my soul. I was but a little boy, it is true, when I was baptized; but this was the influence that came upon me, and I know that it was from God, and was and ever has been a living witness to me of my acceptance of the Lord." (Conference Reports, Apr. 1898, p. 65)
Mark E. Petersen
"That birth of the spirit means something more than most of us normally realize. Through proper teaching, a conviction is born in our soul. Faith develops. Through it we see how important it is to become like Christ. We see ourselves as we are in contrast to a Christlike soul. A desire for a change-over is born within us. The change-over begins. We call it repentance. Through our faith and as part of our conversion or change from one state to another, we begin to see sin in its true light. ... We strive with all our souls to become like the Savior. (Address to seminary and institute of religion personnel, BYU, July 11, 1956.)" (Daniel Ludlow, A Companion to Your Study of the Book of Mormon, p.117)