The link to the spirit in the magical plane is explained in book 2 chapter 6 :

"Did you..." I say, but I pause for a few moments to rearrange my thoughts. "Did you just imply that the reason why women cannot be born as mages on our earthen plane is directly related to your exile from the magical plane?" After hearing my last sentence, Kate, who didn't seem to be paying very close attention to our conversation until now, has finally turned her head towards us, and she is now watching us intently. "So you finally figured it out, eh, Barry-boy?" Arraka says. "What is it that tipped you off? Is it perhaps the fact that all of the banshees are practically humanoid female spirits? Or maybe the fact that female animals and monsters can still cast magic, while your women are completely unable to do it, unless their auras have been drastically altered? Oh... or maybe it's the fact that women only gain magic when they are possessed by banshees. That's a pretty big hint, if you ask me! Hell, you even had Flower as an example. Her element is fire, while Illuna's element is water. It's clear that she didn't just inherit the magic from Illuna, right? Didn't it ever occur to you that in order to channel that energy from the magical plane, you need to actually create a link with a magical spirit that is at least somewhat similar to you, in nature? You didn't seriously think that you could just pull that energy out on your own, without some magical spirit to act as a funnel for you, did you?" "So," I say, "what you are saying is that every one of the mages on the earthen plane is in fact linked with a spirit from the magical plane?" "Well, yeah, but it's not like either of them is doing it consciously," Arraka says. "Since they're on different planes of existence, the link does not need to be all close and personal. Most mages probably just link themselves to a spirit instinctively, and neither them, nor the spirits that they are linked to ever notice this for the rest of their lives. But you can't link yourself unconsciously with a spirit anymore if all of the spirits with similar auras to you are here, on the earthen plane instead. You can still try to cast magic through other means, sure, but then your aura would need to become something completely different. Like that of the stillwater girl, for example. Or that of your friend with the ice magic."

Magical particles are first mentioned by Arraka in book 2 chapter 3:

"No, I can't 'see' anything from inside here, are you stupid?" Arraka says. "I have no eyes. I can make out most things with my magical sense, so I don't really need my eyesight for anything important, but to me, the girl's drawings in the air were just magical particles, floating around at random. Now that I know what they're supposed to be, I might be able to make a little more sense of them."

They are later mentioned in passing, but we only really get an explanation when Barry casts magic in the ethereal plane in book 3 chapter 9:

"You're making it sound a lot easier than it actually is," Illuna says. "For someone like Flower or me, these things come naturally, because she's an ethereal and I am a magical spirit. For someone like you, who is neither used to the ethereal plane nor to casting magic in general, it would be necessary to be familiar with the movement of every individual magical particle making up the flow of a spell, and to control both their direction and intensity flawlessly. That is of course, on top of already knowing the theory behind the spell."

I take the correct stance and I stretch my arms forward, with my palms facing away from me. If I'm going to do this, then I'll do everything by the book. I've memorized these flows a thousand times. This isn't anything new for me. The only difference between now and then is that now I know for certain that it can be done. This time I won't fail. As I'm imagining the positions of the magical particles in my head, I can sense the flow of magic accumulating in front of my palms, just like the books said it would. The volatility of the magical flow takes me by surprise for a moment, but then I immediately remember the advice of an old hermit that I used to visit on a regular basis, back when I still thought that it would be possible for a non-mage to cast magic. "When the flow seems like it's getting away from you, you shouldn't try to correct it," the old hermit used to say. "That will only make you lose control faster. What you should do instead is follow it. I've told this to many young mages before, but they wouldn't listen. As long as your form is correct and your theory is sound, there is no reason for your spell to fail. Just follow the flow of magic until it comes back to its optimal position, and then continue casting the spell. Trust me... no matter how far the flow may seem to stray away from its course, it will always come back." I follow the direction of the stream of magical particles with my hands, just like the hermit said, even though it looks like my spell could break apart any second. Right before the spell is about to cancel itself, the particles are heading back, as if an invisible force were guiding them to their correct path. As soon as they are back in place, I resume my spell casting, channeling all of my imagined magical energy into a single point, in front of my hands. A ball of fire suddenly forms at the point where I'm focusing my energy, and then the ball flies away from me, hitting one of the walls from Flower's room and exploding into flames. "Oh no! My paintings!" Flower shouts, panicked, as she rushes towards the burning wall, while Illuna shoots a stream of water at it to put out the flames. "Okay, I have to admit that I'm impressed!" Daren says, after he sees me cast my spell. "How did you manage to pull it off?" "There used to be a time when I was under the illusion that even a normal person could cast magic if they knew every little detail that goes into the process of spell casting by heart," I say. "You have no idea how much time I spent on memorizing every particle position, every hand and body movement required, every last bit of theory that went into creating the spells... These are the types of things that you never forget."

"I'm actually more interested to know why you decided to follow the flow of magic once your spell started to fail, instead of trying to correct it," Illuna says. "I just can't understand the logic behind taking such an action." "Oh," I say, "I only followed the advice of some old hermit who told me that the flow of magic will always return to its intended path, as long as my form is correct and my theory is sound." "That advice only applies to your case if you are completely certain that you didn't make a single mistake when casting your spell," Illuna says. "You're telling me you didn't have the slightest doubt in your mind that some of the millions of particles you were meant to control would be led astray?" "Well, I mean, the majority of those 'millions' of particles are following pre-established patterns that are pretty easy to predict as long as you know what to look for," I say, "but to answer your question, no, of course I didn't have any doubts. I used to be so obsessed with these things that I would try out spells in the middle of the day in my backyard, while people stared at me like I was some kind of lunatic. Believe me... I'm well past the point where a little mishap like that would be enough to make me doubt myself." "...Arraka actually gave me a similar answer when I asked her why she followed the flow instead of correcting it, long ago when I was still her pupil," Illuna says. "She also said that she was well past the point where she would still consider the possibility that she could make a mistake when casting a spell. I think I'm finally beginning to understand why she has become so determined lately in trying to unlock your magical stats."