The Quick Save State combines the ease of use of the automatic save state with the extra control afforded by manual save states. If you don't see this, you may have to Reset warnings.
The next time you run the game, you will have the option of restarting from the automatic save state or not. OpenEmu automatically saves your progress as you play a game, so that you can always pick up where you left off without having to manually save. You may use a game's built-in save mechanism instead of or in addition to save states, if you like. You can create any number of save states you like, as frequently as you like, and loading them will take you back to precisely where you were in the game when they were taken even in the middle of a battle or cut scene.
Save states differ from the save functionality built into most games - because they are produced by the emulator itself, they carry none of the limitations that the game or console may otherwise impose. Save states are 'snapshots' of a game's progress.