When I lived in an area with soft water, I had the same problem - when I filled a new tank, the water would get cloudy in a day or two, and it took a couple of weeks until it cleared up. And I would get debris settling out, like in your photo. I have NO idea what causes this, but it isn't harmful, as far as I could tell. I used to siphon out the "silt" and go ahead and put animals in the tank when it was still slightly cloudy and nothing bad happened. Interestingly, doing big water changes on established tanks never resulted in cloudiness. And if I followed the advice for speed-cycling (setting up a new tank using a filter and substrate from an established tank) I also did not get this cloudy silty stuff.
If your tap water is rain water, it shouldn't contain anything nasty that would affect axolotls (unless it would come from metal pipes, or something growing in the water in the holding tank). Is this water safe for human drinking?
The problem with rain water for axolotls is that it is probably too soft (not enough dissolved minerals). You might want to think about adding something to add minerals to the water - either aquarium additives, or some form of calcium carbonate.