This unfortunatly is not something I know too much about. I did a bit of research on the subject, and it seems that a major reason that this disclaimer is made is because people who were overweight or obese were not included in the clinical trials for these methods (1). I also turned up some of the actual studies that found that weight is a factor in the effectiveness of birth control (2). However, these studies point to issues with just a couple of methods- mainly oral contraceptives. It is also worth noting that even though these methods may be less effective in overweight people, they are still extremely effective in general (3).
Its easy to get freaked out about all of the noise about birth control and weight, but don’t panic! The best person to talk to is your medical provider. They will be able to recommend the method that will work best for your body and lifestyle.