It saves your game automatically, and whenever you are in the Pause menu you can also save it.
Some of the places that i recall it autosave is when you get a Golden Banana and when you get into the Change Kong Barrel.
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I quote this guy from Gamefaqs
1) I collected a golden banana to see if the game save occurred from collecting a golden banana
> shut the power off and the game saved
2) I then tested to see if the game save occurred from entering a menu
> shut the power off and the game saved
3) Next, I tested to see if the game save occurred when you collect colored bananas
> I collected five colored bananas in fungi forest, shut the power off and the game saved
4) Decided to take it a step further and tested to see if the game saved from progress, no menus accessed, and no items collected at all.
> I ran around the island a few times for about 2 minutes and the game saved my game time. Therefore, the game saved once again from doing practically nothing.
5) I thought to myself, "what the heck is the point of this save feature if the game auto saves", then decided to try one last test!
> I COLLECTED ONE COLORED BANANA, SHUT THE POWER OFF AND THE GAME DIDN'T SAVE!?