Braid's fifth world, known as Time and Decision, introduced a mechanic in which rewinding time created a shadow version of yourself that repeated the actions you had just done. The Company of Myself is a bit like that.
The puzzles here, as they were in that Braid world, are based around performing multiple actions simultaneously. Where it differs from Braid is that here you can create many, many shadows selves, not just the one. You can see three shadow selves in the pic there.
I won't give away much more than that. The pleasure is in discovering it all for yourself. Enjoy.
The Company of Myself [Kongregate]