As a rigorous and comprehensive foundation for electromagnetic information theory (EIT), we present a theory that elucidates the stochastic structure of radiated electromagnetic (EM) fields and induced currents in general EM information transmission systems, encompassing arbitrary random scatterers and EM mutual coupling. The EM system is modeled as a multiply connected, fully general Riemannian manifold with an arbitrary metric. Our approach leverages the exact Green's function (GF) method to construct a novel electromagnetic random field theory (EM-RFT). The GF, interpreted as a response function on the manifold, enables us to treat the internal physical details of the EM system as a black box, shifting the analytical focus toward external input-output relations-a perspective aligned with signal processing and communication theory. This integration of random fields (RFs), electromagnetics, and GFs provides a robust framework for deriving and characterizing the universal stochastic structures of arbitrary EM information transmission systems. Specifically, we rigorously demonstrate the construction of EM random fields that fully comply with Maxwell's equations using the GFs of systems driven by external information fields. Our theory decouples stochastic input RFs from the random fluctuations inherent to the communication system, such as scatterers, and establishes general stochastic correlation propagators for generic EM communication links. Through the Karhunen-Loève expansion, we represent all EM random fields as sums of random variables, offering a foundation for simulating arbitrary EM RFs, evaluating mutual information between input and output spatial domains at any chosen locations within the system, and proving general theorems on the subject. Overall, this theory offers a rigorous and unified mathematical and conceptual basis for EIT, laying the groundwork for the development of new numerical methods to simulate and evaluate mutual information and capacity in EM communication systems.