This is because most countries have compatible copyright laws, so if something violates copyright in America, it probably violates copyright in most of the world.

(The fact that most countries have copied America's absurd extensions to copyright terms is a huge problem of its own, but it's a problem with how legislators and governments have operated in setting the laws & treaties locally, and not with the actual application of the law.)

The DMCA is the US arm of the international WIPO.

The international cooperation is in place because, as I said, no one government has sole jurisdiction.