According to the Pathfinder Magic overview, a wizard has to prepare a spell multiple times to cast it multiple times, which make 0 sense to me. I've also read other comments saying otherwise. Can someone give me some verification please?
Yes, wizards need to prepare spells multiple times to cast them multiple times.
From the page you have linked:
(If you've prepared multiple copies of a single spell, you can cast each copy once.)
Think of it this way: When a wizard prepares a spell, it's like he's loading a bullet into the chamber of a gun. The bullet is the spell, and the gun is his mind. He can load silver bullets, or fire bullets, or steel bullets, or whatever other kinds of bullets he has. He can fire the bullets in his gun in any order, but he needs to load two silver bullets if he wants to fire two silver bullets later.
It's the same with spells. You have to prepare a specific spell in a specific slot, and once you cast the spell in a particular slot, that spell can't be cast again unless you have it prepared in a different slot.
In one of the D&D 3.5 books (I think it was the Player's Handbook, but I may be wrong), there was some flavour text explaining how this worked in-world. Basically, each spell was cast in two parts; a long part and a short part. The long part was cast while preparing spells. This part requires more time and concentration, and once completed, you had a partially-finished spell hanging around until you finished casting it. The short part is what you do to cast the spell mechanically. For example, the standard action used to cast fireball would be the short part of that spell. The idea was that a wizard had to cast the longer part of each spell beforehand, and could only have a limited number of partially-cast spells hanging around at a time. Since casting a spell uses up the prepared part, casting multiple copies of a spell in a day requires preparing the spell multiple times.
If you really don't like this rule, it can be houseruled out pretty easily. A version that I've had some success with is to have wizards prepare a smaller number of spells than usual, but to allow them to cast those spells spontaneously. The way I would run it is as follows:
For each spell level other than 0, wizards can prepare a number of spells equal to their normal number of spells per day, minus 1, minimum 1. The number of spells that can be prepared does not take into account a high intelligence bonus. For example, a 9th level wizard can prepare 1 5th level spell, 1 4th level spell, 2 3rd level spells, 3 2nd level spells, and 3 1st level spells. Wizards can, once per day, spend 15 minutes meditating and studying their spellbook to change out 1 prepared spell per level. The number of spells that they can cast is unaffected. Wizards can cast any spell that they have prepared any number of times, subject to their spells per day limits. This does not count as spontaneous spellcasting for the purposes of metamagic feats, or other features that provide benefits to spontaneous spellcasters. If a wizard wants to cast a metamagic version of a spell, the spell must be prepared that way.
My experience with this ruleset makes me believe that it's pretty fair. The wizard loses a bit of it's minute-to-minute versatility, since they can only get up to 3 spells of a particular level at a time, but they keep most of their power, and get the benefit of being able to cast like a spontaneous caster, for the most part. I'm sure this is terribly abusable in a high-op game, but it's never been a problem for me.
Yes. In Pathfinder (and D&D 3.x) spells are prepared in a specific spell slot, and that slot is expended when the spell is cast. Cantrips are the exception to this; you can cast a cantrip as often as you want.
Makes no sense? Yes, that's a big complaint about the D&D magic system that goes back all the way to the 1980s at least. But it's how D&D has historically done things (though 4e, Pathfinder, 5e, spontaneous casters in 3.x, and psionics before that are handling it differently). Some people want this fixed because it makes no sense, others want to keep it because it's D&D and it works.
There are ways to explain this to have it make sense. For example, a prepared spell could be partially cast during preparation so you only need a quick command to finish it and cast it during combat.
In the old D&D editions, it's mentioned somewhere you need an enchanted charcoal to cast a FireBall. If you use that component previously made, you cannot cast another fireball because you are out of enchanted charcoals.
Consider that wizards, just like any other being, have a limited amount of magical energy inside them and they can divide that between many different spells but all of them have to be previously prepared so the wizards can cast them. Use your imagination to describe how each spells is prepared but consider that the "resource" is expended after the magic is used. Also consider that once the magic is used, the magical energy is freed and can return to the caster but he will need time to prepare new spells.