(Important bit is right at the bottom, but I'll say it here for emphasis):
Only Wizards can use Wizards' spellbooks to cast spells. If you need any in-game reason for that, I'm asking you if you also need an in-game reason why gravity works the way it does, or why Potions only hold up to 3rd level spells, or why Monks are weaker than many other classes.
I can't see anything about that in the PHB but it kinda makes sense...
I don't really understand why that is (except to prevent everyone from carrying around a book of scrolls as my player intends)
The magical writing explain how to draw in the energy (whatever it is) and direct it in the form of the spell (a projectile in the case of Magic Missile and Fireball), but if you have deciphered the Wizards writing then you can perform the somatic, verbal and, if necessary, material components...
The magical writing doesn't do any of these. Where do you get the idea? Or maybe it does, but there's no proof. There's a bunch of things a spellbook might be imagined to do, but the rules give us no clues as to what that might me in a concrete sense.
Maybe this: It's the wizard who fills in the blanks after making sure (a.k.a. memorizing) how he personally will go about manipulating the arcane power flows, which are the only thing described in the spellbook.
Or: A highly complex drawing/diagram/mandala/table contains a plethora of information that make up certain aspects of a spell, in a highly symbolic, encrypted form. The wizard has to memorize every facet of the drawing/etc., after which he is able to draw on the memory. When he does so, and without his doing anything else, cosmic forces make him perform gestures and say words, and an effect comes into being.
Or: the spellbooks contains shorthand versions of stuff a wizard learned when he frequented the academy, made up in a format that is very personal to him, and him only. Using these shorthand reminders, bits of innuendo, nudges in the right direction, a wizard can puzzle out a little facet of how the universe works, but the knowledge is so hard to grasp (let alone without proper wizard training!) that it's highly unstable, vanishing from his mind the moment he tries to get a concrete grasp of what it means (i.e., casts a spell).
why can't you use the Spellbook to cast spells (providing you have appropriate spell slots left)
Rules-y answer: Because you're not a Wizard. End of story.
In-game answer: make one up. It doesn't really matter. As long as somebody ain't a wizard, they can't use a spellbook to cast spells. If they are wizards, they can only use a spellbook to cast spells they could cast within the rules (appropriate level, Int score, not from a banned school etc.).
