Spells are entities from a world parallel to ours. Some call them demons or angels, others the more neutral spirits or gods, but their names don't matter much. The mortal mind isn't made to directly interact with such beings, which is why most of us never realise our numerous encounters with them. This also means that, in order to contact these spirits to coerce, beg or trick them into affecting the physical realm we have to change our state of mind.
The wizards want us to believe they rearrange their mental faculties through study alone, despite their avid use of narcotics being an open secret. It is the reason all wizards seem a bit loony, their minds are permanently at least partially warped so as to be able to access spells. For those doubting the need for stimulants to reach this altered state I would ask: if study is enough why aren't all scholars wizards?
Clerics reach this altered state through fanatic devotion. It is why there is no such thing as a moderate cleric: their fervent disposition is a requirement for their access to the power they call divine. The rituals function as a way to stir up their already excited minds to a point where they are completely out of touch with reality. And ofcourse, it helps that a lot of rituals involve opiates of some kind.
Some minds are born askew. These folk have access to this other realm as naturally as most of us have to the physical. Such people go by many names: Lunatics, sourcerers, the dim witted, oracles and whatever other name there exists for a person who talks about things most of us can't see. They are a rarity, though that might be in part due to the reaction to their existence, both in this world as in the other.
And finally there are those who are chosen. They didn't have innate affiliation with the magical plane, nor did they seek access to it. Instead, something from there has touched their mind, warped it to the point where it can commune with you. They are proof that our minds align more closely with the other world in our dreams, as this is where these beings tend to first appear to their victims.
Thoughts
Animism also lends itself particularly well to magical items. I love how items become magical as described in the blogpost Magic items are born not made by Ben Laurance, and an animist world would allow that to make sense very easily. Especially if you couple it with a belief that the ainu people traditionally have (at least according to wikipedia), that if an item is made with special care that it is endowed with a spirit of its own. A bit tangential perhaps, but I wanted to mention magic items and didn't think it was a big enough idea to warrant its own post.