Somehow means "in some unknown way" or "by some means." When you watch a superhero movie, you know that the heroine will somehow escape the trap set by the evil villain — you're just not sure exactly how she'll do it.
Your best friend might not notice your new haircut, but he still might remark, "You look different somehow." When you use this adverb in this particular way, it means "in an unknown way or for an unknown reason." So your pal knows you changed something, but he can't quite figure out what it is. When somehow first appeared in print, around 1660, it was in the phrase "somehow or other."