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.