So, I have been reading Aristotle and Aquinas for quite a while, but their nuanced terms make their philosophy hardly graspable. It is not clear to me how do they solve mind-body and form-matter problem? Certainly, they divide the world into two principles — potentiality (matter) and actuality — (form), whereas matter cannot exist without a form. But they say that form is immaterial and matter is material, then, it is unclear to me how do they interact, because they say that form organizes the matter? Elsewhere, Aquinas says that form is in all parts of the whole as a final cause, thus unites parts into that whole.
(Note: Your best bet would be to get Edward Feser's Philosophy of Mind where he presents how hylomorphism deals with the mind-body problem.)
There was no such problem for thinkers like Aquinas or Aristotle because hylomorphism does not impose mind-body dualism, but use the concept of soul, instead. They don't conclude that soul is a separate substance like Descartes did, which led him to dualism. Rather, hylomorphism states the soul to be a particular form of the living body such as humans, which involves intellect. Neither form nor matter are substances on their own; Aquinas stresses the form constituting the act–the actuality of the substance and matter–its potencies. So, for Aquinas psychology doesn't study the mind as something distinct, as is the case in Cartesian dualism, but it studies the soul as the form of the body, meaning that it studies what makes organisms whole living things. What we call the mind today is studied by Thomists as an embodied aspect of the organism.
It's worth having in mind that there is no ghost, nor any lifeforce which is the case in vitalism. Aquinas and Aristotle give us a simple statement that a soul is a form of a living thing, and don't have any problems with it.
Intellect is a key aspect of the human soul as opposed to animals or plants, whose souls lack intellect. Intellect differs from sensations in kind. For Aquinas, while senses are grasping the singular entities, the intellect is irreducible to the sensation as it grasps and analyzes the universals. Through senses, we perceive individual things, but intellect grasps say 'roundness', or 'catness'. Senses are limited to material things where intellect knows all things immaterial such as Truth, Good etc. Intellect grasps abstractions that are not physical or corporeal things. Yet, abstract concepts are different from mental images. Sensations give rise to the imagination; mental images of visual, auditory perceptions ("phantasms"). Mental images are always particular as a particular triangle differs from triangularity.
As such the operation of the intellect has the sense-origin rooted in particulars. Because the soul is united with the corporeal body, intellect cannot understand anything particularly, other than by retorting to the mental images (phantasms). Even though intellect is different from imagination and sensation, they all form a trinity through which intellect acquires its content material.
There is also passive and active intellect. Active intellect makes mental images viable for concept abstraction (i.e. deer species). Passive intellect acts like storage that stores abstract concepts for reuse. Epistemologically, this is a kind of indirect realism. Yet, it is very different from other conceptions of indirect realism.
The Cartesian/Lockean conception of mind indicates representations as subjective qualities and gives rise to the problem of the knowledge of the physical world. Namely, how can we ever know anything external that lives outside of representations? How do things point beyond themselves?
For Aristotle and Aquinas though, intellect grasps the forms of things, not some subjective qualities. Form exists both in the intellect and the thing itself. Now, looking at isosceles, the form of triangularity exists both in your intellect and the isosceles. There is just one thing, and not two; the form. It exists in an intentional way (through intellect), and is instantiated in the matter.
Therefore, intellect is key. Intellect is a human's own power to think, it is an active, inherent and embodied quality of humans. It's the activity of thinking which is inherent in the substance of the soul alone. And what is the source of such intellect? it's God-given, through natural law.