These men bear strong and serious witnesses of Christ. That is what they are "called" to be: special witnesses of Christ. Many embrace these men as their spiritual leaders, their guides to Christ and to God.
2017-05-04 UPDATE:My current feeling is that they are mostly well-meaning men, teaching and reinforcing a narrative that is powerful and works for many. I no longer feel the Church narrative is "true," but it certainly is powerful for those that embrace it and take it seriously.
Henry B. Eyring, April 2013 General Conference, "Come Unto Me"
Boyd K. Packer, April 2014 General Conference, "The Witness"
Jeffrey R. Holland, April 2013 General Conference, "Lord, I Believe"
Here's another one from Elder Richard G. Scott:
May the Lord strengthen your resolve, your exercise of faith, and your use of growing character so that you may be the instrument of good He wants you to be. I testify that He lives. As you worthily seek His help, He will guide you in your life. I testify of that with every capacity that I possess. Jesus Christ lives. He guides His work on earth. As apostles of the Lord Jesus Christ, we have experiences that are very sacred that allow us to bear our testimony of His name and of His power. I do that with deep conviction. Jesus Christ loves you. He will guide you in your life. In times of great challenge, when you are perplexed about which turn to take, kneel down and ask your Father in Heaven to bless you, and let your faith in the Savior and His Atonement be the rock and foundation of a successful life, in the name of Jesus Christ, amen.