The Boy Scouts of America's website states:
The Boy Scouts of America is one of the nation's largest and most prominent values-based youth development organizations. The BSA provides a program for young people that builds character, trains them in the responsibilities of participating citizenship, and develops personal fitness.
In addition, the BSA requires all members to espouse a belief in some "higher power", in other words, whoever you believe to be god.
The 13th Article of Faith of the LDS Church states:
We believe in being honest, true, chaste, benevolent, virtuous, and in doing good to all men; indeed, we may say that we follow the admonition of Paul—We believe all things, we hope all things, we have endured many things, and hope to be able to endure all things. If there is anything virtuous, lovely, or of good report or praiseworthy, we seek after these things.
Clearly both organizations are seeking after character-filled and moral members. Specifically, the BSA's Scout Law states:
A Scout is:
Trustworthy, Loyal, Helpful,
Friendly, Courteous, Kind,
Obedient, Cheerful, Thrifty,
Brave, Clean, Reverent
(bolded entries emphasizing those that appear in the Article of Faith in some form)
As an Eagle Scout myself, and having got to know many Mormon Boy Scouts, I can attest that many are there because they believe in the principles of both organizations. The strong overlap between the two sets of values is likely what makes Scouting so appealing to Mormon young men.
The Church's statement on Scouting:
Where Scouting is authorized by the Church, quorums may participate in Scouting activities during Mutual. Under priesthood leadership, Scouting can complement the efforts of Aaronic Priesthood quorums and Primary classes in building testimonies in young men and boys.”
For more resources, you could check out the LDS-BSA Relationships website. As for the LDS Church's affiliation with scouting, it is the largest chartering organization (sponsor for local troop), with 37,856 units containing 430,557 youth, almost 70,000 more than the next-largest organization.