How do the Ninjago spinners work?
I've had a few of these in the past, and I've managed to find one so hopefully I can explain it with a few pictures.

They are simple, hand-powered spinning tops, best spun by a finger and thumb:


Usually the original spinner would have a Plate, Round 2 x 2 with Rounded Bottom on the bottom to aid spin:


Although clearly these became more advanced as per the other examples with pin holes in the bottom for further attachments.

The minifig stands inside the spinner, only lightly held on by the parts inside:



The interesting part is the light grey part that holds the minifig's feet which can go up and down, which happens when the dark grey part is rotated:
Lowered:

Raised:


And it was this action that made the game you'd play with them: By getting two people with spinners, you could spin them at each other, and hope that they'd bump, or at least the characters with their weapons would catch on each other, and this would cause them to rotate within the spinner, lifting them up, and hopefully knocking one off!


The cards would then allow you to add additional weapons to your minifig, or parts to the spinner itself, either raising it up, or building up "defences" that would stop the minifig from rotating, and so keep it in place for the next battle.