Is it actually true that birth control will help with body hair ?

Yes, often: Excess body hair in areas affected by sex hormones (face, upper back, upper abdomen, breast/chest) increases with higher levels of androgens ("male" hormones). The ocp suppresses their production, increases the blood binding protein (shbg) and thus the amount of free, active hormone. Slowly, the excess hair thins and stops increasing. Other methods also help cosmetically.