The oscillation instability of sessile drops is ubiquitous in surface acoustic wave (SAW)-powered digital microfluidics. Yet, the physics underlying these phenomena has not been elucidated owing to the complex interplay between hydrodynamics, acoustics and capillarity. We decipher the instability by combining high-speed imaging of the droplet surface vibration with inner acoustic pressure measurements. We rationalize the observed behaviour with a model inspired from optomechanics, which couples an intracavity acoustic mode excited by the SAW to a surface deformation eigenmode through amplitude modulation and delayed radiation pressure feedback.