The present theoretical research addresses the problem of the subgap excitations detected in the electromagnetic response of superconductors with strong microscopical disorder. Such materials have recently attracted considerable attention thanks to a wide variety of their practical applications, ranging from single-photon detectors to qubits. However, all such applications substantially rely on the absence of the aforementioned dissipative modes at low-energies, but proper theoretical description of these excitations has not yet been introduced. In the present work, we use the Anderson pseudospin model on a random regular graph proposed in previous studies to suggest a candidate for a physical mechanism behind the low-energy excitations. We start by introducing a theoretical approach that provides full access to statistics and spatial structure of the order parameter in a strongly disordered superconductor away from the superconducting transition. Using the obtained results, we then develop a comprehensive theory of phase fluctuations of the order parameter. The results of our analysis are backed by numerical experiments. Our findings suggest that within a certain range of parameters the modes of phase fluctuations can indeed be found at arbitrarily low frequencies while the global superconducting order is still present in the system.