Depending upon your church background, the sight of a minister wearing a robe may be unusual, even unsettling. Why do we do it?
1. Wearing a Robe hides the man
The fact is that we are often distracted by what the pastor is wearing, especially if it is out of accord with the church tradition we grew up in. If the minster wears an expensive suit, people will notice and wonder why the pastor has spent so much money on his clothes. If the pastor wears a cheap suit, people will notice and wonder why the pastor doesn't dress more nicely. If the pastor wears a hawaiian shirt, people will be appalled at the lack of formality...you get the point. The simple black ministerial gown is designed to remove these barriers and detract attention from the man and his taste in clothing, to whom he represents. Which leads to the second point.
2. Wearing a Robe emphasizes the office
The minister wears a robe for the same reason that a judge wears a robe in court, or a policeman wears a uniform when on duty - to identify the fact that they are acting in official capacities. Likewise, the minister wears a robe to signify that when he is conducting the worship service, he is acting in his official capacity as a man who has been called by Christ to proclaim his word to his people. The robe reminds the congregation (and the minister!) that he is in the pulpit as an ambassador of Christ, and thus is not to speak his own thoughts or words of wisdom, but only the Word of God.
As a brief historical note, although the practice of protestant ministers wearing robes is not widespread today, it used to be. Until the middle of the 19th century most protestant ministers wore robes for the above reasons. If you want to know why the practice was in large part abandoned, click here.