Complex sentence: Henry Jerome was a band leader at the Hotel Edison who, after hearing about them from Bill Randle, signed Dorsey Burnette, Johnny Burnett and Paul Burlison to a management contract.
Split into simpler sentences: Henry Jerome was a band leader at the hotel when he heard from Bill Randle about a trio. This led the signing of Dorsey Burnette, Johnny Burnett and Paul Burlison to a management contract.