When Calpurnia is asked why she speaks like other people of color when she attends church, her explanation is that she does not want her fellow congregation members thinking that she is condescending to them.
She even tries to frame her answer in a way that Scout can understand, by asking her if she would feel "right" about using a different dialect in the presence of her family and friends. By putting Scout into her own (Calpurnia's) shoes, Cal helps Scout to understand why her "double life" is necessary.
The irony is that Calpurnia is capable of not only speaking correct, unaccented English, but she is also one of the few church members who can read and write, by her own admission. Calpurnia's ability to slip between the worlds of white and black southerners is a bit distressing to the children at first, but once Cal calmly explains its necessity, the scene is one of the more memorable in this novel.
One "language" is her formal speaking and one is her casual speaking voice. Think about how you talk to your close friends (or text) and then think about how you would talk to your parent's boss--one is more formal and the other is less. She doesn't technically speak two languages--they're both English.