I did some searching of past talks to find out what prophets have said regarding this notion.
The single biggest evidence I found came from President Benson.
In his talk "The Keystone of our Religion" President Benson mentioned three reasons for why we should make the study of the Book of Mormon a life-long pursuit.
His sentiments, with regard to the Book of Mormon being written for our day, have been echoed by other General Authorities. The main point is that we need to learn from the Nephites' mistakes. We can read about how the Nephites fell because of pride and materialism and how they defeated their enemies. We would be wise to take to heart the lessons the Nephites struggled to learn.
The second great reason why we must make the Book of Mormon a center focus of study is that it was written for our day. The Nephites never had the book; neither did the Lamanites of ancient times. It was meant for us. Mormon wrote near the end of the Nephite civilization. Under the inspiration of God, who sees all things from the beginning, he abridged centuries of records, choosing the stories, speeches, and events that would be most helpful to us.
Each of the major writers of the Book of Mormon testified that he wrote for future generations. Nephi said: “The Lord God promised unto me that these things which I write shall be kept and preserved, and handed down unto my seed, from generation to generation.” (2 Ne. 25:21.) His brother Jacob, who succeeded him, wrote similar words: “For [Nephi] said that the history of his people should be engraven upon his other plates, and that I should preserve these plates and hand them down unto my seed, from generation to generation.” (Jacob 1:3.) Enos and Jarom both indicated that they too were writing not for their own people but for future generations. (See Enos 1:15–16; Jarom 1:2.)
Mormon himself said, “Yea, I speak unto you, ye remnant of the house of Israel.” (Morm. 7:1.) And Moroni, the last of the inspired writers, actually saw our day and time. “Behold,” he said, “the Lord hath shown unto me great and marvelous things concerning that which must shortly come, at that day when these things shall come forth among you.
“Behold, I speak unto you as if ye were present, and yet ye are not. But behold, Jesus Christ hath shown you unto me, and I know your doing.” (Morm. 8:34–35.)
If they saw our day, and chose those things which would be of greatest worth to us, is not that how we should study the Book of Mormon? We should constantly ask ourselves, “Why did the Lord inspire Mormon (or Moroni or Alma) to include that in his record? What lesson can I learn from that to help me live in this day and age?”
And there is example after example of how that question will be answered. For example, in the Book of Mormon we find a pattern for preparing for the Second Coming. A major portion of the book centers on the few decades just prior to Christ’s coming to America. By careful study of that time period, we can determine why some were destroyed in the terrible judgments that preceded His coming and what brought others to stand at the temple in the land of Bountiful and thrust their hands into the wounds of His hands and feet.
From the Book of Mormon we learn how disciples of Christ live in times of war. From the Book of Mormon we see the evils of secret combinations portrayed in graphic and chilling reality. In the Book of Mormon we find lessons for dealing with persecution and apostasy. We learn much about how to do missionary work. And more than anywhere else, we see in the Book of Mormon the dangers of materialism and setting our hearts on the things of the world. Can anyone doubt that this book was meant for us and that in it we find great power, great comfort, and great protection?
As the cliché goes, those who fail to learn history are doomed to repeat it.