You ask:
What explanation systems do humans use (not just in philosophy)?
All domains of discourse use logical inference to predict events and states. Natural language contains logical operations inside of informal logic to make predictions about the world around us. This constant stream of inferences might be seen as inferentialism. While not all explanation (SEP) is temporally predictive, explanation is inherently about preserving truth when inferring conclusions from premises.
While many people recognize mathematical explanation (SEP) and scientific explanation (SEP) as important form of explanation, even small conversations and attempts to predict the future function as explanations when externalized. In this language, one might see the brain as a control system that attempts to predict future events based on certain and probabilistic reasoning. Such views give rise to models of cognition such as the Bayesian Models of Cognition (mit.edu).
All in all, there are an unlimited number of forms of explanation, because argument is fundamentally a generation of explanation. Each argument can be located in a different domain of expertise. Therefore a firefighter might provide explanations about combustion. An electrician can provide explanations about routing power and current. A teacher provides pedagogical explanations. All of these are part of the broader attempt to persuade people using logos. For an in depth analysis of how argument is performed in informal logic, see Toulmin's model of argument articulated in Uses of Argument.