Answer: It sounds as though your plants are developin a fungal disease commonly called damping off. Although they get off to a good start, your seedlings are either getting too much water or not enough air circulation to prevent this disease.
Start all over again with fresh seed and new seed starting mix or potting soil. After planting your seeds, keep the soil moist but not soggy wet. When they germinate, try to water from the bottom rather than the top (to keep the foliage dry) and be sure there is good air circulation all around. I set up a small fan to help keep the air moving. Don't aim the fan on the plants, but set it so it directs the air either in front or behind the tray or pots of seedlings. Water your seedlings only enough to keep the soil from drying out and pulling away from the edges of the pots or trays. I think you'll have better success with your next batch of seeds.
Enjoy your new plants!
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