This a fungal disease called "damping-off" and is the most common disease that occurs when you start plants from seeds. Seeds sown indoors and ones planted outside are both susceptible. The seeds germinate and look healthy but suddenly wilt and die for no reason. Several fungi can cause this including species of Pythium, Rhizoctonia, Botrytis, and Phytophora, which are all found in the soil. These plant pathogens can also be found surviving on potting benches, seed containers, planting tools and on the floor.
There are several ways that damping-off can attack. This pathogen can strike before the seeds germinate causing them to become dark and mushy preventing them from germinating. When damping-off strikes the shoots of new seedlings, the stems shrivel and the seedlings fall over and die. Seedlings that are a few days old are usually able to resist damping-off.
Conditions that favor damping-off are high humidity, overcrowding of seedlings, not enough air circulation and over watering. Keep seedlings thinned out to avoid overcrowding and improve air circulation. You may also try placing a small fan next to your seedlings. Try to keep the area relatively cool (60-65), since tomato seedling grow strong at this temp, but the fungus likes it warmer. Make sure there is adequate drainage and water is not standing in the bottom of the flat. Once the seeds have sprouted remove any cover to allow the soil to dry out. This pathogen is present in the soil so it is important to keep your garden tools and seed flats or pots clean. Wash them in warm soapy water rinse in a solution of 1 part bleach to 9 parts of water and then rinse again with clean water. Always use sterile soil-less growing medium and never use soil from your garden unless it has been sterilized. If you want to sterilize your soil, pour boiling water through the soil and let it drain freely. To be 100% sure, you can bake soil in the oven for 30 minutes at 180-200 degrees, or in microwave on the highest setting for 15 minutes. Beware, this is not a pleasant smell so make sure you can do this on a day when the windows can be opened.