Complex sentence: They began playing at the Peppermint Lounge, soon becoming the house band there, but left in April 1965 to become the house band at Sybil Burton's nightclub, Arthur.
Split into simpler sentences: They began playing at the Peppermint Lounge, soon becoming the house band there. They left in April 1965 to become the house band at Sybil Burton's club, Arthur.