So I bought Silent Storm last night, played for a couple hours and got my characters to level 4 and quit for the night at the base. Today I loaded my save and played one mission. About 10 minutes into the mission I noticed all my characters had gained 6 levels. All I had done in that mission was sneak around behind the enemies that were distracted by a bunch of my allies. 6 levels in 10 minutes or so seems really quick to me and I was wondering if there was some kind of xp bug or other explanation for this as I feel like I really didn't earn those levels. Thanks.

Sounds like a bug for sure.
On my pure vanilla SS (straight off GOG, no mods) the lead character in my team is a Scout and her Spot skill increased very rapidly as in one point every few minutes. All I did was set to Hide mode, spot an enemy and leave her there to spot for a few mins.

You can modify your autoexec.cfg inside the cfg folder.
This allows you to use the console and from there you can roll back the 6 levels. The game will reset your characters skill to match a Level 'x' soldier. All soldiers will get affected, skills may be reduced if they are deemed too high for that level.

Full instructions and other console commands here
Please backup all your savegames and autoexec.cfg prior to doing the change.
I've used the command to increase level before but not rollback, so im not sure if there are side effects, like if you've spent the ability points gained; then when you decrease levels what is going to happen to those points that you no longer 'have' but already allocated.

http://au.ign.com/cheats/games/silent-storm-sentinels-pc-619907

This happened to me as well, but after a random encounter instead of a mission. I don't know if it was just that the encounter was incredibly difficult, but it was still too big an increase in levels.

Thanks for the answers! Glad to know someone else has had the same kind of thing happen. I think I'm just going to use the save from right before the fight and do it over.