[The Law of Common Consent] is that in God’s earthly kingdom, the King counsels what should be done, but then he allows his subjects to accept or reject his proposals. Unless the principle of free agency is operated in righteousness men do not progress to ultimate salvation in the heavenly kingdom hereafter. Accordingly, church officers are selected by the spirit of revelation in those appointed to choose them, but before the officers may serve in their positions, they must receive a formal sustaining vote of the people over whom they are to preside.1
Elder Loren C. Dunn explained the responsibilities that accompany the sustaining process (this answers why it is important):
When we sustain officers, we are given the opportunity of sustaining those whom the Lord has already called by revelation. … The Lord, then, gives us the opportunity to sustain the action of a divine calling and in effect express ourselves if for any reason we may feel otherwise. To sustain is to make the action binding on ourselves to support those people whom we have sustained. When a person goes through the sacred act of raising his arm to the square, he should remember, with soberness, that which he has done and commence to act in harmony with his sustaining vote both in public and in private
Another good article on this subject is The Law of Common Consent, by Matthew O. Richardson. While it may not be official LDS doctrine it is well written with several citations to scriptures, prophets, and history of this practice.