Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.