It has a unique saving system.
For a temporary save, such as if you're in the middle of a dungeon and want to stop and return later, you activate a owl statue (There are several throughout the world) and can return later to that exact time and state.
For a permanent save, you have to sing the Song of Time. That will rewind you back to the start of the first day and the major items you own will still be in your inventory. But it resets the world, you lose your rupees (You can deposit them at the bank before doing the song if you want to save them), you lose your arrows, you lose your bombs, etc.
So basically to save your game, you use the owl statues if you're in the middle of something important (Such as clearing a dungeon or one of the many sidequest, some of which can be pretty involving). And you use the Song of Time at the completion of something major like clearing a dungeon to save your game before proceeding to your next task you want to tackle.
The Nintendo 64 emulator doesn't have the suspend state feature that many of the Virtual Console emulators have. So don't exit via the home menu thinking that you can return to where you left off the next time you load the game. Only the original save system of the Nintendo 64 release of Majora's Mask is present here.
Check out the included mini manual via the home button. I believe that also covered the save system if you're still sort of confused. It's hard to quite grasp it until you actually experience it.