The jackpot machine is entered through the checkpoint (starpost). You need 20 rings to 35 rings to enter. If you have this amount of rings and you hit the checkpoint, you would see orange stars encircling the starpost. Jump into the stars to enter the jackpot machine!

(Note: You can enter the slot as many times as you want)
Additonal information :
Here is a list of what you would get if you get three of a perticular object.
3 Sonic = 30 rings
3 Knuckles = 25 rings
3 Tails = 20 rings
3 Robotnik = -100 rings (I repeat, this is minus 100 rings!)
3 ring = 10 rings
3 bar = 8 rings
3 jackpots = 100 rings