It's all about adventure -- specifically, about a game's ability to put players in an adventure, making them feel like they're influencing a whole fictional world with their actions. Which is something that wasn't particularly common in shoot-em-ups from the early '90s.
In his newest video, Really Freakin' Clever argues that Star Fox for the SNES achieves this sense of adventure with excellent level design, and by making the player actually feel something by telling its story through its gameplay. Take a look:
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