To me, especially with an app with a touch interface, the borders serve a clear purpose of defining the area that is able to be used to touch/click to trigger an action.
Take the pink quiver icon. Without that border, the apparent touch area of that icon would be small, even if the actual touch area was larger, and it might give a user pause as they try to position their finger in order to hit a small icon area that doesn't actually exist.
Of course, this only really starts to matter if your icons are actually thin or small, and thus cover only a small area. And in all such cases, there's no reason the border/background couldn't be a very subtle one anyway.