**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.