The gartel holds no clothing up. The coat it is tied over is already closed, and the belt adds nothing structural, which is what makes it worth explaining. The reason given in the sources is separation: a division at the waist between the upper part of the body and the lower, so that the heart and the mind are set apart from what lies beneath them during prayer.
The requirement behind it is older than Hasidism and is not confined to it. The codes require a separation between the heart and the lower body for prayer, and require preparation before standing before God, taking up the prophetic verse about preparing to meet your God. Most Jews satisfy that with an ordinary belt or with the waistband of trousers. The gartel makes the same requirement visible and deliberate, and the act of tying it is the boundary between what a man was doing a moment ago and the prayer he is about to begin.
That is the sense in which it is an act of attention rather than an item of dress. A man putting on a gartel is doing something rather than wearing something, and the deliberateness is the content. The word itself is Yiddish for belt, from the same root as the English girdle, and the object is generally a woven cord or a flat band of silk or wool, most often black.