That is a pretty short scripture, but it packs a powerful punch. It also goes against the worldly view of a leader. The dictionary says that a leader is "the person who leads or commands a group, organization, or country." I think the Savior is saying that a leader should serve those that he leads.
In the Church of Jesus Christ of Latter-day Saints, we have Bishops that are called to lead a congregation of members. By the dictionary definition of leader, our bishops would lead or command our congregations, but in my experience, that is not the case.
LDS bishops are not compensated in any way and most bishops are anticipated to fulfill their duties for at least 5 years. I'm not sure there are many people with great leadership aspirations that would willingly give up their free time without any compensation in worldly terms. The work of a bishop is not glamorous and they don't get public accolades like the leaders of companies or political leaders.
As members, we always refer to the work a bishop does as his service. This is what the Savior was talking about. A bishop serves his ward and its members. He serves by mourning with those that mourn and comforting those in need of comfort. He helps those that are in need of physical support. He is charged with helping the members spiritually too. Doesn't really sound like a leader who commands, does it?
The Savior taught that when we are put in a position of leadership, we need to serve those we lead. He was the Son of God, the greatest of us all, and He was the perfect example of a leader who served. He healed the sick, gave sight to the blind, forgave the sinner, and lifted the hearts of the lowest in society. He gave his life for all of us as His great final act of service in this mortal life. I know that the reason so many follow Jesus Christ, is because through His service to all mankind, He leads us back to the presence of God.