Shortly after the release of their landmark 1979 LP London Calling, The Clash found themselves being referred to in marketing and by fans all over as "The Only Band That Matters". This moniker attaches itself well to a band that made protest and social justice a major focus of their lyrical messages, so it has stuck fairly well all these years later.
According to a 2013 interview with CNN, Mick Jones said, "the record company came up with that; it wasn't us. We weren't like trying to blow our own trumpets. It was kind of a moniker and it stuck. Every group wants to be the best group when you're young."
CBS Records, the label that The Clash were signed to throughout their run as a group, definitely used the slogan abundantly to help sell the band even harder. Musician Gary Lucas has claimed to be the originator of the phrase at the record label, as he was working in their creative services department at that time.