Instructions
Get to high ground. Despite their spider-like appearance, Spinners are unable to climb, so if you can get above them you will be a lot safer.
Disable the Spinner using paint thinner. Tap the "Z" button to splash green paint thinner on the Spinner, or, hold down the "Z" button to spray a stream of thinner. The paint thinner erases the Spinner's shell and flips in on its back rendering it temporarily disabled.
Kill the Spinner using the spin attack. Approach the disabled Spinner and shake the Wii remote to use the spin attack. The spinner won't stay on its back for long, so you need to move fast to kill it. If the Spinner gets back on its legs, you will have to disable it again using thinner.