You twist off the "sprouts" when they are about 6-8 inches long and plant them in the ground. They are called 'slips' and will root in the ground and you should get sweet potatoes in about 3-4 months. Most of the time they like poor soil.

Cheers,
DL

Here is a link that might be useful: How to grow sweet potatoes - some videos

I got a 'crop' of about 30 large tubers from 2 plants. The vines spread in my front yard [no lawn] and passers-by commented on the beautiful 'ground cover'. I started my plants by the above method. Now I just take cuttings from the vines, keep them in my bedroom window [in water] over cold season [I don't think we have a real 'Winter' here in SoCalif.]They grow thick roots, and transplant out in the garden like champs!
This year I'm also planting in a side yard that gets very little watering, because SP are said to be 'drought tollerant' [we're on drought water costs, so cut way back on watering some parts of my yard]

Mine have done best when grown in a ridge of soil about 6-8" deep so the SP can expand easily as they grow. Our soil is a clay loam with lots of compost/rotted manure added.

Buy sweet potatoes from organic food store.Lay them horizontal
at the top surface in pot, pot filled with potting soil.Cover the potatoes with potting soil to keep them even with soil surface.Water regularly.In about 10 to 15 days you will see the sprouts.Separate the sprouts when they reach about 3 inches in height and plant them in 6 to 8 inches of hills.You can harvest 10 to 15 slips from a single potato depending upon the size of the potato.If you need more slips fix more potatoes in the soil.