Because an adherent is usually something or someone that sticks to something or someone else, it's logical that it can also mean a person who believes in and supports a certain philosophy or leader.
An adherent to a cause can be called a disciple, one who believes and tries to get others to believe. It can also refer to a material that sticks to other things. While the first meaning comes from the 15th century, the reference to a sticky substance didn't emerge until the 16th. In all cases, the word comes from the Latin root haerēre "stick," connected to the prefix ad- "to," making the word mean "to stick to."