Why does iPhone Voice Control turn on when I wear headphones while working out?
It certainly sounds like the headphones are causing this. It's likely that moisture is eventually getting into the in-line switch/microphone module which is bridging the switch terminals and sending spurious signals to the phone.

However, I'd try the following:


Check for liquid damage inside the headphone socket: Hold your phone up to the light so that you can see down inside of the headphone port. There's a small indicator at the bottom of the port which will have turned a red/pink colour if liquid has come into contact with it. If positive, this could indicate that the phone's headphone socket may (also) be at fault.
Obtain another pair of the same Apple headphones and see if the issue occurs again.