In this paper we prove the convergence of the finite volume MultiPoint Flux Approximation (MPFA) O scheme for anisotropic and heterogeneous diffusion problems. Our framework is based on a discrete hybrid variational formulation which generalizes the usual construction of the MPFA O scheme. The well-posedness and convergence of the scheme is derived assuming a local coercivity condition which can be easily checked numerically. The novel feature of our framework is that it holds for general polygonal and polyhedral meshes as well as for $L^{\infty}$ diffusion coefficients, which is essential in many practical applications.