Spore like fungi/mold in seedling tray?

Hi all, newbie here with my first post!

I started some Cayenne Pepper seeds on my window cill last week.  Before the seedlings appeared there were lots of thin white spindly spore like strings which started shooting out from the soil in various directions.  Most of the seedlings have now broken the soil and seem to be growing fine and the stringy things seem to be dying off, but I'm worried about what they are and if they will a) cause harm to the plant or b) cause harm to me if my seedlings are successful and I get some peppers to eat.

I looked up damping off disease but I notice this looks more like cotton wool, which I also have a little off, so am wondering if they are connected?

Any help is greatly appreciated!  Thanks in advance!

Gemma