The story
This phrase comes from when radio DJs played vinyl records. The disks had two sides - on a 45 RPM disk, sides A & B. The song that the record company wanted to promote most heavily would go on the A side. So when a DJ played that side, they'd sometimes say, "And now on the flip side...", and play side B, the lesser known track.
Etymology from Wiktionary, CC BY-SA 4.0.