How do you treat a sprained or torn arch on the bottom of your foot?

R.I.C.E. If all you have done is sprain it (get it checked out by a professional to rule out a break, dislocation, or major tendon tear), then the basic principles of r.I.C.E will help out. Rest, ice, compression, and elevation. Stay off your foot for several days-to-weeks, ice the area for 10 min every hour to relieve pain and swelling, use an ace wrap for stability and to reduce swelling, and elevate it.
See a podiatrist. You might need x-rays to rule out injury to the bones or joints, and if the sprain or tear is significant enough you might need to limit activities, wear a brace/splint, take antiinflammatory medicine among other things.