The red exclamation points indicate broken data paths. It's looking for data (feature classes, shapefiles, or whatever) on that computer, and can't find them, either because they aren't there or the file path has changed.
Make sure all the files associated with the document are transferred on to the other computer, and when they are, make sure the file path is the same. Or, you can re-set the paths by clicking on the red exclamation points (I think, I know this is how it works in Arcmap, not sure about explorer.