Answer: There are a number of reasons for why this might happen, but the most common are poor air circulation possibly combined with insufficient light, and either overwatering or possibly underwatering. First of all, though, you should be aware that the first "seed" leaves, or cotyledons, will die off as the "true leaves" grow on. This much leaf loss is normal.
Once the seeds germinate you need to remove the plastic cover right away and place the plants in either a bright window or under lights or even outdoors in a very sheltered spot in part shade (if the weather is warm). This should ensure adequate light levels.
If several seeds germinated in each cell, you will need to remove all but one of the baby plants. In general, the idea is to keep the sturdiest looking one. The best way I have found to do this is by cutting the extras off at the soil line with a pair of tiny sharp scissors. Some gardeners simply pull them out, but that method risks disturbing the roots of the plant you wish to keep. (Some gardeners will also "prick out" a few and transplant them if they need more plants.) Between removing the cover and thinning the plants you should improve air circulation and maximize the available light.
Watering new seedlings is a little bit tricky because it calls for some judgement. You need to keep the soil moist but not soggy. At the beginning, the roots don't reach very far down so the top level of soil must be kept moist. As the seedlings get bigger and the roots go deeper, you can allow the surface of the soil to dry a bit between waterings. When the plants have filled the cells with roots, you need to be sure they don't dry out too fast -- the larger plants use more water than the small plants.
In my experience, I have found that tiny new seedlings watered from the bottom tend to stay too soggy, so I will water once or twice a day gently from the top. Some gardeners mist their seedlings to water them and some simply pour slowly and carefully. Do keep in mind that misting provides a gentle watering but it can spread and/or encourage foliar problems. In any case, the frequency will depend on how much air circulation there is and on how warm it is. (Moving air and higher temperatures cause faster evaporation.)
Later on, as the plants get bigger I may switch to bottom watering. If you use this technique, be sure to pour any excess water out of the tray as soon as you see that moisture has wicked up to the soil surface.
Good luck with your seedlings!
Q&A Library Searching Tips