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Abstract Living cells can adapt their shape in response to their environment, a process driven by the interaction between their flexible membrane and the activity of the underlying cytoskeleton. However, the precise physical mechanisms of this coupling remain unclear. Here we show how cytoskeletal forces acting on a biomimetic membrane affect its deformations. Using a minimal cell model that consists of an active network of microtubules and molecular motors encapsulated inside lipid vesicles, we observe large shape fluctuations and travelling membrane deformations. Quantitative analysis of membrane and microtubule dynamics demonstrates how active forces set the temporal scale of vesicle fluctuations, giving rise to fluctuation spectra that differ in both their spatial and temporal decays from their counterparts in thermal equilibrium. Using simulations, we extend the classical framework of membrane fluctuations to active cytoskeleton-driven vesicles, demonstrating how correlated activity governs membrane dynamics and the roles of confinement, membrane material properties and cytoskeletal forces. Our findings provide a quantitative foundation for understanding the shape-morphing abilities of living cells.