Complex sentence: Whatever the case, the release of Shaw's recording of "Beguine" skyrocketed him and his band to fame and popularity exceedingly fast and the recording became one of the most famous and popular anthems of the entire Swing Era.
Split into simpler sentences: Whatever the case, the release of Shaw's recording of "Beguine" skyrocketed him and his band to fame and popularity exceedingly fast. The recording, indeed, became one of the most famous and popular anthems of the entire Swing Era.