The nature of the union of mind and body is causal. There is a two-way interaction. When I place my hand on a hot plate, my hand (or hand plus nervous system, etc.) causes my mind to have the thought "Pain!" Meanwhile, my mind causes my hand (or nervous system plus hand, etc.) to pull away from the hot plate and put ice on it, or whatever.
Descartes says that the point of causal interaction is the pineal gland, which is between the two hemispheres of the brain: " a certain very small gland situated in the middle of the brain's substance and above the passage through which the spirits in the brain's anterior cavities communicate with those in the posterior cavities" (Passions of the Soul, CSM I, article 31).
Very fine particles of matter, animal spirits, flow underneath the pineal gland. Somehow these animal spirits imprint an image on the pineal gland. What the mind can do, apparently, is inspect this image, and, through the pineal gland, which functions a bit like a weather-vane, 'sway' these particles this way or that way. This changes the direction of the flow of particles. This has the result of making the body do this or that.
Descartes does not explain how it is that a mind, an immaterial substance, not in space or time, can influence a body in this way.
Descartes does not explain why it is that a mind can only affect one body (indeed, only one part of a body) and not any other body (or indeed, any other part of the body).
Descartes does not explain why the mind cannot causally interact with other minds, but only with one body, whereas the body can causally interact with other bodies. (Note, he does accept that God, an infinite mind or infinite thinking substance, can causally interact with (a) all other minds, and (b) all bodies, if He so wishes. But he has no explanation for why only God can do this).
What is called psycho-physical causation is left unexplained. However, what Descartes argues is that we experience mind-body causal interaction at every moment. When we are injured, we feel pain. When we want to do something, our body moves. It is an ordinary fact of life. Since it occurs, it is of course possible for it to occur (anything that is actual is possible, or, if something does happen, then it is true that it can happen).
His argument might be as follows:
(1) A mind and a body are distinct substances.
(2) I am a mind.
(3) I have a body.
(4) I experience the thoughts of my mind causing the movements of my body.
(5) I experience the movements of my body causing the thoughts of my mind.
---> My mind and my body are distinct substances that causally interact.
If this argument is valid, then it remains to determine whether it is sound, that is, whether all of the premises are true. Premise (1) is a conclusion reached on the basis already in Meditation VI. It is the conclusion reached in the Argument for the Distinction between Mind and Body (Cartesian Substance Dualism). That argument, however, may be understood as merely hypothetical. It says that minds and bodies are distinct, without saying that there are any minds.
Premise (2) however asserts that I am a mind. I am a non-extended, indivisible, thinking substance. Descartes doesn't prove this. He needs to prove that thought, which is non-extended and indivisible, cannot arise from a substance that is extended and divisible. Since premises (4) and (5) essentially rely upon the previous ones, the contentious premises are (2) and (3).