[China, Vietnam, Malaysia, Brunei, the Philippines, and Taiwan] each claim parts of the territory based on what is known as UNCLOS, or the United Nations Convention on the Law of the Sea, which says nations can claim an Exclusive Economic Zone extending 200 nautical miles into the waters surrounding their coastlines. They may also make further claims based on the continental shelf extending from their shores.
Since a lot of natural resources lie under the sea bed, controlling the seas above those resources--which the Law of the Sea allows, is now important. Important enough to go to war over.
Remember, as I've noted before, LOST says nothing about how to determine ownership of the islands that form the basis of those 200 nautical mile economic zones.
The UN. Is there anything it can't make worse?