Labor unions are able to increase the wages of their workers because they can, in a sense, gain monopoly power in a given firm and because they can restrict the supply of labor. When a union can restrict the supply of labor and a firm cannot buy its labor from a substitute, the union is able to gain higher wages for its workers.
In some cases (though not as often as in past times) unions control all the jobs of a certain type in a company. Only members of the union can hold these jobs. This means the union is the only seller of labor to that particular company. When the union is the only seller, it can restrict the supply of labor. It can say, for example, that its members will only work X hours per day. It can say that workers doing job Y will not also be able to perform tasks associated with job Z. In these ways, the union can reduce the supply of labor.
Supply and demand analysis, then, shows us that unions can increase wages for their members. They reduce supply and they prevent companies from buying their labor from outside the union (making demand for union labor less elastic). These factors allow unions to charge higher wages for their workers.