The irony of that statement, is that the PRC (and ROC to a degree) is reasserting claims made and seems to be seeking the balance of influence China had at various points under imperial rule.To the Chinese; what they're doing it only to be expected given its new status as a world power. The see it as their right. They're only doing now what in the past they were unable to do. To them, outside powers which object are only being hypocritical given their past track record and that as outside powers, they have no business interfering in issues China has with other countries; in an area the Chinese consider their backyard. over matter concerning Chinese ''sovereignty''.
It is nowhere near the point of modern Italy laying claim to most of Western Europe, North Africa, the Balkans, Asia Minor, or areas containing modern Syria, Lebanon and Israel, but that should illustrate my point.
At various points in Chinese history, Imperial China was arguably the most powerful nation in the world, with influence and control which spanned a vast area for the time. For a variety of reasons, that balance of power (both in absolute and comparative degrees) has waned. My take as an outsider is that the PRC is seeking to regain some of that comparative power, having observed other nations (which did not even exist when China was at it's height) which had gained power and now hold it. Further (and again, this is my take on things only), it seems as though some of the areas which had once been under the control and/or influence of China have a degree of 'cultural memory' which is causing them to be less than eager for China to regain some of that influence.
If one looks at two of the current Great Powers, which in the past had vastly greater territory and comparative power, namely the UK/British Empire, and Russia/Soviet Union/Tsarist Russia, one can see two very different attitudes held by many of the nations which had made the respective powers. I tend to think the attitude of former subject-nations towards China/the PRC has more similarities with the attitude towards Russia than that of the UK. With that in mind, and with the belief that should the PRC claims be recognized it would be at the expense of rival SCS claimants, I can see how a number of ASEAN member would be concerned.
For nations/powers outside of the region like the US the SCS is still a vitally important maritime area simply due to the volume of global trade (~40% IIRC) which passes through it, as well as the direct impact on the economies and resulting imports/exports of major trading partners like S. Korea and Japan. Should the PRC be able to control the SCS, it would have the potential ability to extract a 'toll' from trade to/from S. Korea and/or Japan, or force shipping bound for those countries to be re-routed further east through the Philippine Sea, taking a longer route and likely increasing expenses due to more fuel being required and delaying deliveries and pickups of imports and exports.