Meaning of "within 10 days from when they move"

In this context,

From: indicating the point in time at which a particular process, event or activity starts

• The show will run from 10 am to 2 pm.

So, you need to let them know of your address change within 10 days after you change it. (Although, as Em1 said, if you know about it beforehand, there is no harm in letting them know earlier.)

It's poor grammar and it seems that it could actually be either as it could mean 10 days from (as in before) the move date or 10 days from (as in since) the move date, but common sense says it probably means "10 days AFTER the date you move" as this probably makes the most sense anyway - folks might not know their address until after they move, if they are still finding a place. You can't make "knowledge of future residence" a requirement of anything like deposit, etc. so it may not "mean" much - are they going to take you to court otherwise?