There's really two key points to remember - your face and your shoes.
The first time I went to try something on and they handed me one of these face covers, I had no clue what to do with it. I think my first thought was "maybe this is a bag, perhaps something to put items I want to purchase in?"
I was the epitome of a confused foreigner but after some puzzled looks the shop attendant gestured to me that it was a face cover, and the manner in which I should be putting it on. When you think about it, the idea is genius really - there's been so many times when I have been back home trying clothes on, and seen that the garment has foundation smeared all over the neckline from whomever had tried the item on before me - not exactly what you want when you're buying a cute new piece of clothing, right? This eliminates that problem completely!
Usually what happens is one of two things. The shop attendant will either hand you one of those face covers, if it's the kind of store that has a shop attendant walk you to the fitting room with your items (or that has someone waiting outside the fitting rooms). Otherwise, you'll usually find a box of them in the fitting room itself - just pop it on your head with the longer, veil type part covering the entirety of your face. Some added bonuses of these are that your makeup isn't going to get all smudged, and for some reason those masks tend to minimize the static frizz in my hair that comes with trying on multiple tops, at least for me! Winner!
When it comes to shoes, it's customary to take them off before going into a fitting room in Japan. You'll see in the picture below that the fitting rooms are ones that you have to step up into - take your shoes off before stepping up, and leave them outside like the sneakers you can see in the picture.
Some fitting rooms aren't like this one, where you need to actually step up into it - they're the type which have a carpeted floor section inside - just take your shoes off before you step onto the carpeted portion, and place them neatly on the outer tiled section.
Then, other places like my local H&M, don't seem to strictly enforce the shoes rule and I've seen people going into the fitting rooms with shoes on, but since living here I err on the "when in doubt, take the shoes off!" to save any potential awkward situations arising.
I actually love the concept though of taking shoes off now - I think it just keeps things so much cleaner, and it's nice to walk into a perfectly tidy fitting room without mud or other gunk tracked in by people! We're a "no shoes inside" household now and it definitely makes things a lot easier when it comes to keeping the floors clean at home, too!
Hopefully this brief summary spares you some of the struggle that I had when first moving here and not knowing exactly what to do - particularly with those face veils!