You can start by reading the article: The Divine Institution of Marriage
The article illustrates very clearly why marriage between a man and a woman should be preserved and protected. The primary purpose of marriage is to nurture children and teach them to be responsible, moral adults. "Extensive studies have shown that in general a husband and wife united in a loving, committed marriage provide the optimal environment for children to be protected, nurtured, and raised."
The article concludes with the following:
"Strong, stable families, headed by a father and mother, are the anchor of civilized society. When marriage is undermined by gender confusion and by distortions of its God-given meaning, the rising generation of children and youth will find it increasingly difficult to develop their natural identity as a man or a woman. Some will find it more difficult to engage in wholesome courtships, form stable marriages, and raise yet another generation imbued with moral strength and purpose.
"The Church of Jesus Christ of Latter-day Saints has chosen to become involved, along with many other churches, organizations, and individuals, in defending the sanctity of marriage between a man and a woman because it is a compelling moral issue of profound importance to our religion and to the future of our society.
"The final line in the Proclamation on the Family is an admonition to the world from the First Presidency and the Quorum of the Twelve: “We call upon responsible citizens and officers of government everywhere to promote those measures designed to maintain and strengthen the family as the fundamental unit of society.” This is the course charted by Church leaders, and it is the only course of safety for the Church and for the nation."