This is a way of asking “How are you?” using the verb to fare. In this sentence, faring means more-or-less the same as doing in “How are you doing?”
But it has a more literary style and therefore more serious connotations. To fare is an old verb in English, which originally meant to journey or to travel. For example, one of the oldest poems in English is called The Seafarer—one who journeys by sea. So, “How are you faring?” sounds a little poetic; it really means “How is your journey (through life) going?”
You can use this in formal contexts, and also between friends. It implies more concern than “How are you doing?” For example, if you talk to a friend who has recently suffered a loss or hurt—such as death, divorce, job-loss, or injury—you might ask “How are you faring?” to mean “How are you managing?” or “How are you getting by? Are you okay?”