@Aminey My understanding is that the spirits are just nature itself coalesced into dense forms, probably with magic as a core and also possibly as a binding agent. Think of the spirits as pearls. Pearls are made when clams form keratin around an irritant with the intent of isolating it to prevent further irritation. Magic would be the irritant, and a specific element would be the coating. Alternatively, you could also compare them to planets, where elements condense around a single point that is pulling them in due to some sort of gravity, magic likely being the "core" and the "gravity" within this scenario. The Magic stones would then be spirits that have become so dense they materialize, either due to them literally collecting too much magic and that of it's corresponding element(similar to how some stars can turn into black holes when they collect too much matter) or due to them actually "dying," with death either being a time limit(a life expectancy) or them simply running out of magic which is used to keep them in one piece. My honest bet is that most of them just condense like black holes, which would suggest that the bigger the magic stone the bigger and more powerful the spirit would have been before "death," but there could easily be some other reason as to why they die. Furthermore, the planet idea would also explain why anyone can use magic at all. Specific elemental spirits are drawn to those with similar souls, such as fire spirits drawn to someone who's soul has the essence of fire. Just so happens that the nobility have particularly strong essences, hence why commoners aren't typically able to use magic or at least potent magic because they're essence isn't strong enough to attract spirits.
Ultimately, this is all just my assumption. I assume much of this will be revealed later in the series, but I guess it also depends on how much depth the author wants this to have.