I think that that statement is confusing "citizen" and "national". There are 6 types of British nationals (some can have more than one): British citizen, British overseas territories citizen, British overseas citizen, British subject, British National (Overseas), and British protected person. They all hold British passports, but the latter 5 statuses listed above do not generally (there are some exceptions) confer right of abode in the UK. "British citizens" always have right of abode in the UK, so the statement is redundant; but I think they were probably trying to say that holders of British passports are not necessarily able to use the Visa Waiver Program, and incorrectly used the term "British citizen".

Another example where nationals don't have right of abode is Republic of China (ROC) nationals, where the nationality itself is not enough to have right of abode in ROC-controlled territories (Taiwan and some other islands). You also need Taiwan household registration to have right of abode  in those areas. Technically, most people in Mainland China, Hong Kong, and Macau, as well as many people of Chinese descent abroad, are also ROC nationals, but they do not have Taiwan household registration (and cannot easily get it) and thus cannot live in Taiwan. Children born abroad to people with Taiwan household registration also don't automatically have Taiwan household registration, but it is easy for them to establish Taiwan household registration once back in Taiwan. ROC nationals without household registration can get ROC passports (except those from Mainland China, Hong Kong, or Macau), so an ROC passport does not denote right of abode in ROC-controlled territories. Whether an ROC passport holder has Taiwan household registration is indicated by the presence of a Taiwan ID number in the passport. Only ROC passports with a Taiwan ID number are eligible to use the Visa Waiver Program.