Most importantly, they have felt the witness of the Spirit often and strongly enough to be able to say that they know Jesus is the Messiah. Surely others can say the same thing, but these people have been called and ordained specifically with the obligation and authority to bear that witness.
As important as the witness of the Twelve is, however, it isn't the only work they are called to do. When Jesus ordained his apostles (Jn. 15:16), he also made them a special administrative council (Lk. 6:13; in Acts, Luke portrays them as such a council). Similarly, the second highest council in The Church of Jesus Christ of Latter-day Saints is that of the Twelve Apostles. Immediately above them is the First Presidency: the Prophet and his counselors. (The senior Apostle becomes the new prophet at the death of the previous prophet; his counselors are usually chosen from among the apostles.)
In their capacity as the second highest level of church administration, the Twelve oversee the administration of the LDS Church as a whole. For that work these men's backgrounds in business, science, education, and the law is at least as important as theological training might be.
The Twelve Apostles of The Church of Jesus Christ of Latter-day Saints are good, prayerful, and faithful men. They have served for years in a variety of capacities within the Church, coming to know members of all sorts, from different cultures and countries, with different backgrounds, needs, and expectations. That service has prepared them well for their prophetic role of knowing and preaching the gospel of Jesus Christ.
Like the ancient Church, the LDS Church expects its everyday affairs ultimately to be governed by apostles: the daily affairs of the Church should be governed by those with a knowledge of Christ. Mormon apostles' education, with their work and community experience outside the Church, has prepared them to administer the Church's programs in worldwide, complex setting.
But the most important part of what they do as apostles is the testimony that they bear of the reality that Jesus is Christ, the Messiah. His resurrection was the first testimony to the world of his messiahship. The apostles' testimonies spread that first and fundamental testimony. Beyond whatever else the Church might need contemporary apostles to do, that witness to the Church and to the world is their first responsibility.