Hello Garaleos,
thanks for your feedback! Clerics use the prayer book to store the necessary words for the spell. The prayer book takes up one slot, but can hold the words (prayers, incantations) for many spells. Using a spell causes one Fatigue per spell level. So a cleric with three available (empty) inventory slots could e.g. cast 3 level one spells or 1 level 2 spell plus one level 1 spell. The Fatigue fills the inventory slot until the cleric has taken a rest.
Learning new spells can be narrative driven. E.g. the cleric could study in a temple/monastery to learn new spells. The Cairn B/X document also says: "Clerics can copy new clerical spells from other prayer books"
Hope this helps?