What happens is that Mario is invited to visit Princess Peach at her castle, because she tells him that she has a nice cake baked for him, sadly, when he arrives, he finds that Bowser has taken over the building and trapped Peach its walls. Inside the castle Mario finds that there are numerous paintings which serve as portals to other worlds and, using the castle as a hub, he then sets out into the various different worlds to collect Power Stars so that he can pass through special doors and rescue Peach.
For being the game that laid the groundwork for pretty much every 3D title which came afterward, Super Mario 64 is very impressive. There are some really exciting worlds like the inside of a volcano, a huge underground labyrinth filled with poison gas, an island which changes size and so many more. Everything is so fun to explore and the worlds all feel very open and natural. The boss fights are suddenly much more exciting too, and there are multiple battles with Bowser which are all very fun and memorable.
Furthermore, there's a whole new set of power-ups this time. There's the Wing Cap, which allows Mario to fly through the air for a limited period of time and which really lets you see how open the world is. There's the Metal Cap, which is my personal favourite, and that turns Mario into the indestructible Metal Mario who can also walk around at the bottom of bodies of water. Maybe it's the cool music, maybe it's that Metal Mario looks really cool or maybe it's the fun of being invincible, but I really love the Metal Cap. Finally there's the Vanish Cap and it turns Mario invisible and lets him walk through certain walls (which is nice, but not great).
But while it is certainly a very good game, there are one or two downsides. The first is, that while the worlds are all very nice to explore, they sadly feel a little empty, it would be nice to have seen some of the various citizens of the Mushroom Kingdom and been able to interact with them. As things are, the levels are all populated almost entirely by enemies, which is a bit of a shame. The enemy designs also often look a bit like generic video game enemies and are graphically rather unimpressive, I feel that they sometimes spoil atmosphere. But these are only small problems with what is an otherwise very good game.
(I do not own the copyright of the screen shot.)