I don't see the directory when I start ScummVM. I click on the add game button, and then I tried clicking on the red go up button. I see a note pop up that says go to the previous directory level. I don't see an option that will let me go down to a lower directory level. I thought that clicking on the note would do something, but when I move my mouse it disappears.

   

Which version of ScummVM? Which Android version?

   

I've tried version 1.9.0, the latest development build and the latest stable build. None of them showed a directory when I clicked Add Game. I'm using an Acer Chromebook 14 CB3-431 that has Android 7.1.1.

   

You need to update to 1.9.0.1

   

I did an update. I now have 1.9.0 (June 25, 2017 16:39:20). Is that the right one?

   

I could be mistaken but I've read varied user experiences with Android apps on Chromebooks, some work really well, some crash or just don't work at all. Maybe you're experiencing something in between where the directory structure is not being recognized appropriately.
I don't have a Chromebook myelf so I cannot try reproducing it, just my 2 cents...