Yup, I think TZ nailed it (as usual). Damping off is a fungal condition that little seedlings are very vulnerable to. It is seen mostly in conditions of too much humidity/moisture and too little air circulation.
The peat pots are very bad for this, hold too much moisture in (and then when they finally do dry out, they suck water away from the seedlings)
What you need is plastic pots (can be plastic drink cups with holes in the bottom for drainage, yoghurt/ sour cream cups or whatever you have like that). And you need trays to put them in so you can water from the bottom up. Just put a little
water in the bottom of the tray and let the plants soak it up. Two other tips: A little personal fan to circulate air helps. Putting some chamomile tea and A LITTLE bit of cinnamon in the water you water with helps (they are natural anti-fungals).
But the plants that are wilted and lying on the soil are goners; nothing unfortunately you can do to save them, sorry. I hate how they hype those peat pots; it's hard to find anything else in the garden stores these days and they are plant killers. How do they get away with that!
If I were a little more paranoid, I'd think it's a conspiracy to make people think it is really hard to start things from seed, so they will keep buying lots of plants from H.D and places like that.