Zurg, with Agent Z's assistance, stole and corrupted the Uni-mind with evil at one point, planning to use it as the power source of a giant machine designed to have the Uni-mind link the minds of all the galaxy's inhabitants to his evil.
Because many LGMs are Star Command's scientists and engineers, the organization reacted badly to the loss of the Uni-Mind and sought to attack Planet Z to bring it back, but were unfortunately additional targets of Zurg's mind control.
Buzz, Mira, Booster and XR, now the only Space Rangers left, managed to slip by Planet Z's defences and turn the Uni-Mind to the good side again, which caused Zurg to lose control of the entire galaxy.
Exactly how the Uni-mind works is uncertain. The Uni-mind only works when kept in the building it is housed in on the LGM Homeworld. Once it is removed from that precise spot (or, more likely, when taken off the planet), the LGMs are no longer connected to it. Thus, it is not necessary to destroy the Uni-mind to break the link between the LGMs.
The reason it must be at that spot is unknown, though it may be because that is where most LGMs are located, and thus that is where their minds would connect. When Zurg removed it from that spot, their minds wouldn't connect.
It would also appear that through the Uni-mind, the LGMs are able to share their knowledge and intelligence, thus their confusion and Star Command's response once the link is broken.
The Evil Emperor Zurg had figured out how to use it to link the minds of others to his own after corrupting it with his evil (how this is possible is never explained, though apparently impaling it with his fingers somehow caused it to link with his mind), but he was foiled by Team Lightyear of Star Command. When corrupted by Zurg, the Uni-mind lets out a noise that sounds eerily like screaming.
The Uni-Mind may also retain small parts of personalities of non-LGM beings that have linked with it. An LGM chanted Buzz's catchphrase unintentionally and floated into the sky following Buzz's purification of the Uni-Mind.