The debug.err file is generated each time you start FE/LH. It records hardware and software configuration info and a log of events for that session of the game. You can find it in your Documents\MyGames\LegendaryHeroes directory. Since it's recreated on each start you'll only find one.
Most of us who have been beta testing for a while use DropBox or a similar service to pass files to the developers. DropBox is a free service that lets you store files on a cloud server; it has a "Public" folder that lets you generate a URL that others can use to see the file. If you install DropBox there will be a DropBox folder on your desktop. Open it, open the Public folder inside it, and copy your debug.err into it (maybe rename the file to something unique in case you want to save more than one). After the file is automatically uploaded to the DropBox server (there will be a green check on the lower left of the file icon when that's complete) right click on the file, and choose the DropBox, Copy Public Link option. Then paste that link into a forum append and the devs will be able to see your file.
Note that on crashes, you'll often also find files named like legendaryheroes_x_crash_2013_....mdmp or .zip that are memory dumps that are also very useful to the developers. You'll find them in Program Files\Steam\SteamApps\common\FE Legendary Heroes.
(Note: There are other cloud services that do the same function as DropBox. I only have experience with DropBox, so that's the one I can explain how to use. You can get DropBox from dropbox.com).
