Computational modeling of cardiac electrophysiology presents significant challenges due to the steep and fast wavefronts in action potential propagation and the coupling between reaction-diffusion equations and systems of ordinary differential equations describing ionic dynamics. High-order discontinuous Galerkin (DG) methods offer attractive features for these problems, including natural handling of complex geometries. However, the development of efficient preconditioners for DG discretizations remains a critical bottleneck, particularly for realistic three-dimensional cardiac geometries. We present a novel agglomeration-based multilevel preconditioner that exploits the flexibility of polytopic DG frameworks to accelerate the convergence of iterative solvers for the monodomain model. Starting from a fine computational mesh, we employ a fully automated, dimension-independent R-tree based spatial indexing algorithm to generate nested hierarchies of coarser polytopic grids through element agglomeration. The key innovation lies in constructing multigrid preconditioners on these agglomerated polytopic grids. The nested approach enables the construction of efficient intergrid transfer operators, implemented in a matrix-free fashion. This contrasts with classical algebraic multigrid (AMG) methods, which struggle with the redundancy of degrees of freedom in DG discretizations, particularly for higher polynomial degrees. Numerical experiments demonstrate the effectiveness of the proposed methodology across multiple scenarios, spanning from two-dimensional benchmarks to three-dimensional realistic left ventricle geometries. The preconditioner exhibits excellent parallel scalability and maintains nearly constant iteration counts with respect to polynomial degree, while classical AMG preconditioners show significant degradation for higher-order elements. The method also shows robustness in iteration count with respect to the number of multigrid levels. The implementation is based on the polyDEAL (https://github.com/fdrmrc/Polydeal) framework, built upon the deal.II (https://dealii.org/) finite element library, natively supporting parallel computation via MPI with efficient load balancing.