The people of Krypton had to live somewhere, but as I got it from Man of Steel the Phantom Zone is just some place in space. But some character (I don't remember Jor-El or Zod) wanted to send people to the Phantom Zone to save them. What is the difference between Phantom Zone and just any other place in space?
I think you might have mixed things up a little bit.
Jor-El (Superman's father) wanted to send the Kryptonian genetic codex (the library of Kryptonian genetic material) to another planet where it could be used to restore Kryptonian society.
Zod, however, rebelled against the council and tried to steal the codex for his own purpose - to remove those bloodlines that he deemed "degenerative" and reform Kryptonian society in his own image. Ultimately he was unsuccessful and arrested after killing Jor-El (who succeeded in launching a ship carrying his son and the codex towards Earth). The council sentenced Zod and his followers to banishment to the Phantom Zone for their crimes as treason.
As for why the Kryptonians use the Phantom Zone as a punishment - I don't think the film makes it particular clear, but from other DC material (comic books and the 1980s movies), it's a "pocket universe" outside of our own where you don't really have a physical presence and it is rather bleak.
Kryptonians abhor capital punishment and will go to incredible lengths to avoid it. Damning convicts to a different dimension may be a cognitive heuristic that allows this universe to remain "clean". For example, objectively and scientifically, we know that evil cannot "infect" a material object... but it's well studied and established that nonetheless we will feel icky wearing an item of clothing said to be Hitler's versus one by a great humanitarian, even if it should make no material or meaningful rational difference. This is a "law of contagion", a mental shortcut, that sentient minds are susceptible to and critically, explicitly called out by Zod in the film! Asked if he has any last words, Zod replies (emphasis added):
You won't kill us yourself! You wouldn't sully your hands! But you'll damn us... to a black hole for eternity!
He follows that calling them a pack of fools. Zod is pointing out the ridiculousness of the ceremony and dimensional exile when the result is the same as "killing us yourself" meaning the only actual difference is the fact that to a Kryptonian it means not "sullying one's hands" (e.g. the "law of contagion").
There are many cultural trappings that aren't fully rational but explained by this kind of idea.
The Phantom Zone was meant as a prison for criminal Kryptonians. That is basically all that needs to really be said but let's get into detail. Jor-El had different views from Zod and decided to take the codex but failed since it was no longer in place. It was with baby Kal-El (Superman A.k.a Clark Kent), who was already sent off Krypton. Zod was sent to the Phantom Zone for treason of trying to overthrow the council. Not to mention his buddies Ursa and Non assisted Zod making them go to the Phantom Zone along with him. (Original Superman) In the Man of Steel reboot version of Superman, Zod had more followers. Once the planet was destroyed, the Phantom Zone prisoners were released. This brings me to the many theories that come from this events. Anyway. Superman defeats them and are commonly said to have been sucked into a black whole.