Yeah, there's two options there. The guy on the left means you can solo fight three enemy soldiers at a time and even if you die horribly in seconds (which can happen if you have a character you're not used to!) you can just try again. Plus you keep all EXP and Phantoma, and you can even do certain Requests in there (eg kill 30 enemies with ice magic) so it's well worth it.

The guy on the right is as LWL says it is, he will offer you "secret" training. This will save your game and boot you to the title screen, and the longer you leave it the more levels your character will have gained. I always make a point of making my way here whenever I'm about to turn it off for the day. Heck, even if it is just for a few hours you can normally gain a level or two. But that's just a neat trick and not the training I was specifically recommending