What explanation systems do humans use (not just in philosophy)?

A useful way to break things down is by forms of reasoning:

• Deductive reasoning: infer the logical implications of some premises or statements.
• Inductive reasoning: make a generalisation or future predictions from some premises or evidence.
• Abductive reasoning: try to figure out the best explanation for the evidence - typically the explanation with the greatest explanatory power.

Maths and logic use deductive reasoning. Though the axioms and premises need to be based on inductive or abductive reasoning - you can't get very far by pure deduction.

Determining what someone would do in some situation could be based on inductive or abductive reasoning.

Theories of gravity, and science in general, is inductive or abductive, not deductive, reasoning.

Alongside and related to this, foundational parts of producing accurate predictions are:

• Empiricism (the scientific method, broadly speaking)
  • You use observations to get a baseline understanding of reality.
  • You take your reasoning and test it in the observable world, to verify that it actually works, and to fix or improve it.
• Consistency and non-contradiction
  • For reasoning to be sound, it should produce the correct results when applied consistently across a number of different questions. If reasoning is uniquely used to try to draw one conclusion, and it applies nowhere else, it's functionally useless and entirely untrustworthy.
  • Reasoning that produces contradictions is not sound.

If you determine what someone else would do based solely on what you'd do, you're often going to be wrong, as different people have different motivations and preferences and ways of thinking, driven by differences in biology and life experiences. By refining this determination through empiricism, one can predict outcomes much more accurately, by considering what you know about differences in how people think, and what you know about the specific person in question, by considering your motivations compared with theirs, etc.

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 forms 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.

Emotionional decision-making followed by post-hoc rationalisation

We do what satisfies our emotional needs, then we make up a story to pretend that we had a rational basis for what we did.

I would say logic and introspection are conscious systems, but the more interior "systems of thought" such as create qualia are unknown. Rather than make things up about what is unknown we can look to the names that have historically been given to what we are describing. That is, what makes a quale is the intelligibility of reality, (which some might say is a quale itself, albeit next-order). As the producer of intelligible reality, what has been said about it? How about Plato, at Republic 508e

what gives the objects of knowledge their truth and the knower's mind the power of knowing is the form of the good.

Or Heraclitus, fragment 1.

all things come to pass in accordance with this λόγος (logos).

Ok, the 'form of the good' and λόγος are pretty vague, but these are the early coinages for what makes appearances if you take it in the sense of intelligible reality and qualia, (and not in the sense of what makes reality in the first place).

'Intelligibility' itself is also one of these vague 'systems of thought', though much later coinage (late 14c.), including as it does the PIE root *leg- (1) "to collect, gather," with derivatives meaning "to speak (to 'pick out words')." — back to logos.

The issues about Scientific Explanation are many.

An explanation must be nomological (involving laws) and causal.

But a deduction from scientific laws is not enough: it must be produced in a relevant context.

More specifically, explanations should be in a certain way "robust": they should apply to a wide range of possible situations and not only the very specific situation that actually occurred.

This is known as rejection of ad hoc hypothesis: "an ad hoc hypothesis is a hypothesis added to a theory in order to save it from being falsified."

Current scientific explanation can provide us answers for only a limited number of questions.

Qualia is a phenomenal term: it describes a sort of fact (old: feeling, sensation) that needs explanation.

For an attempt to a more philosophical answer, we may see: Jonathan Lear, Aristotle the desire to understand, (Cambridge UP, 1988), page 1: Aristotle's Metaphysics begins [980a]:

"All men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness they are loved for themselves; and above all others the sense of sight. For not only with a view to action, but even when we are not going to do anything, we prefer sight to almost everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things."

Aristotle is attributing to us a desire, a force, which urges us on toward knowledge. [...] Aristotle no doubt believed it was this desire that motivated him to do the research and thinking that led to his writing the Metaphysics, and he trusted in this desire to lead others to study it.

And see Met, 993a27: " The investigation of the truth is in one way hard, in another easy."

Thus, we may say that if we read understanding as a process: "the investigation of the truth", then complete understanding is the positive result of this process.

See also Lear, page 6:

Although 'to know' is an adequate translation of the Greek 'eidenai [εἰδέναι],' Aristotle used this term generically to cover various species of knowing. One of the species is 'epistasthai' (literally, to be in a state of having episteme) which has often been translated as 'to know' or 'to have scientific knowledge,' but which ought to be translated as 'to understand.' For Aristotle says that we have episteme of a thing when we know its cause [Post.An, 70b9: "We think we understand a thing whenever we think we are aware both that the explanation because of which the object is is its explanation, and that it is not possible for this to be otherwise."]

To have episteme one must not only know a thing, one must also grasp its cause or explanation. This is to understand it: to know in a deep sense what it is and how it has come to be. Philosophy, says Aristotle, is episteme of the truth.

And see Understanding for an overview of the many-sided issue: Explanation vs Understanding, understanding in the social sciences compared to the natural ones.

Warning: the following is my own original thinking, and although it's inspired by a lot of other philosophers, it's not a part of any philosophical tradition.

Explanations can be divided into causal and non-causal. Causal explanations involve change and time, and the structuring of nature so that one event leads naturally to another. Non-causal is everything else and involves things like mathematical or ethical explanations.

Causal explanations can be divided into teleological and non-teleological. Teleological explanations involve a purpose or goal or (in older terminology) an end, and non-teleological explanations do not.

Teleological explanations can be divided into design explanations and organic explanations. Design explanations are based on the intentional activity of an intelligent agent. Organic explanations are teleological explanations that don't involve the active participation of an agent. They include things like biological growth, development, and death, and also includes the operation of mechanical devices like cars. What I mean is that although a car is the outcome of design, when you are explaining how parts work rather than where it came from, you use organic explanations like "the reason gas is fed into the cylinders is to power the action of the pistons".

Non-teleological explanations can be divided into the mechanistic and the nomological. In mechanistic explanations there is one part of the world called the cause that acts upon another part of the world. The cause may be an object, force, energy, field, or the like. In nomological explanations, the whole world is seen as active together, evolving according to some rule or equation.

Causal explanations can be further subdivided into natural and supernatural. I described the natural versions above, but there are supernatural variations. In the case of design, a supernatural variation would involve supernatural agents like gods or spirits. In the case of non-teleological explanation, it would involve some object outside of time or nature that acts as the cause. For example, if you want to explain the Big Bang without God, you need something outside of nature to explain it, some thing or rule that is outside of nature that causes nature to come into existence.

Causal explanations can alternatively be divided into specific and aggregate causes. For example when you have a bunch of agents all working to their own ends in a society, the outcome may be somewhat predictable but it isn't really designed--that is, there is no mind whose decisions led to that particular outcome. The study of this sort of causality is called economics. There is something similar in a collection of living things that all are working out their own ends but interacting with each other in a way that produces an outcome that is not specifically an end for any organism. The science that studies this is ecology.

Aggregate behavior of non-teleological explanations is called statistical mechanics or a stochastic system, and it can give rise to emergent properties like pressure that aren't the property of any of the individual objects in the system. If you really wanted to maintain the parallels, you could argue that statistical mechanics is the aggregate behavior of a mechanistic system and stochastic systems are the aggregate behavior of nomological systems, but that may be pushing things a bit.