How can you heal a sprained ankle really quickly?

A sprained ankle needs to be "rested". You can elevate it on a pillow and apply ice (20 minutes on... 20 minutes off) to reduce swelling. If given crutches, USE them. Stay off your foot until the time your doctor says and definitely as long as there is pain or swelling. You can resume walking as you are able to bear weight-- usually within a couple weeks.

Take care with strenuous activities or if walking in snow/ice or up/down stairs. Also take care with twisting motions, such as turning corners.

If you follow your doctor's orders, you'll be back on your feet in no time.