In the simple case, explicit reasoning is reasoning which:

Can be explained in language.
Uses well-defined terms.
Relies on well-understood reasoning steps.
Has clear assumptions and conclusions.
You are aware of doing when you do it.

"What do I mean by explicit reason? I don’t refer merely to “System 2”, the brain’s slow, sequential, analytical, fully conscious, and effortful mode of cognition. I refer to the informed application of this type of thinking. Gathering data with real effort to find out, crunching the numbers with a grasp of the math, modeling the world with testable predictions, reflection on your thinking with an awareness of biases. Reason requires good inputs and a lot of effort." - Jacob Falkovich, The Treacherous Path to Rationality

However, the exact definition may vary based on context. For example, explicit reasoning might be operationalized as imaginary verbal reasoning taking place in a person's inner monologue (ie, in auditory working memory). In other cases, we might have a much higher standard, eg actual symbolic logic written on an external medium such as paper. So, reasoning can be more and less explicit, along several dimensions.

Explicit reasoning is one of many modes of reasoning by which humans may reach conclusions. While it is not always the best mode of reasoning, it has the advantage of being scrutable, ie, open to inspection. This makes it easier to correct, in particular through imaginary verbal reasoning modelled after dialogue (ie, mentally responding to yourself as if you were another person, with critiques and corrections). Since it can easily be recorded, it can also be subject to feedback from many other people, which can further improve the quality of this type of reasoning. Also, explicit reasoning can easily be chained together to reach less obvious conclusions.

Other types of reasoning include inner sim, gestalt pattern recognition, mental imagery, and Gendlin's Focusing. Important questions include when to trust different modes of reasoning, how to combine the results of different reasoning modes, and how to best facilitate communication between different modes of reasoning.

Related to: Dual Process Theory