The "Hourglass" in the name is a reference to the fact that by using its active ability, you are granted a few more seconds to live. Within those few seconds, your cooldowns will continue to drop and your team may come to help you.
Champions like Twitch can benefit from these few seconds to "fade to stealth" by stealthing then activating Zhonya's Hourglass. After emerging from the stasis, they will be fully stealthed.
Zhonya's Hourglass can help with taking down turrets as you will lose aggro when you become untargetable, meaning you don't need to waste time walking out of turret range for it to transfer targets, and makes it safer to stay under fire for a longer time.
Activating the active will cancel channeled abilities; however, non-channeled area abilities such as Fiddlesticks's and Morgana's ultimates will continue, making Zhonya's Hourglass particularly powerful when used in the middle of an enemy team, allowing them to continue doing damage while being invulnerable to counterattack.