What stops step data from the accelerometers on the phone and watch from doubling up?

Apple Health is smart. It automatically aggregates step data, so that every minute, it takes the step count from the last 60 seconds from the device with the highest input and adds that to the daily total. It knows when the measurements from each device were started and when they ended, and it makes sure to not include any steps from two sources measured at the same time. This is a post-processing thing, not in the time of measurement: it's not that the Apple Watch shuts off the iPhone's accelerometer; rather, they both generate step data, as in your screenshot, but Health figures out which step counts to use from both in an aggregation-type process after the fact, getting you the most accurate and complete step count possible, while not exaggerating.