Fitness+ is for access to the video programming (i.e. workouts) similar to offerings by other companies (e.g. Equinox, Peloton, etc.). It uses the Fitness app to manage workouts, which is how your health data is generated/tracked/displayed.

You do not need to pay for Fitness+ to record workouts in the Fitness app. However, (without Fitness+) the only other way to start a workout within the Fitness app is to do so from an Apple Watch.

If you'd rather stay completely free, instead use say, Strava or Pedometer+ (or free app of choice) to start a workout (in those apps) and connect the app to Apple Health. Preview your health data in the Health app instead.