Damping off - It's fungal. The plants are unlikely to recover. To prevent it in the future while your plants are still in pots, make sure not to over-water, and possibly run a fan on your plants. Try bottom-watering, to ensure the top-most layer of soil doesn't get wet, as this fungus stays on the surface. The fan will help dry out the top layer quickly, too, and gives the added benefit of strengthening the stems before they go outside. Of course, you don't want to blow your plants away - if you can, point the fan in such a way that it circulates air without being directly on your plants. Otherwise, give it some distance.