The story summary wrongly gives the impression that the US is forever interfering in ICANN's affairs. This is simply not true: while it does retain an oversight function, it has never used that to interfere in ICANN's operational matters. It does have ultimate oversight authority, and so is in theory the final recourse if ICANN should go off the rails. The question is, if the US gives that up, who gets the final say?
ICANN is a body that has power that Slashdotters should care about. It writes rules into the contracts it has with top level domain Registries, rules that individual domain registrants must obey. Mostly these rules are technical not policy, but that is changing. ICANN has long required domain registrants to submit to ICANN's Uniform Dispute Resolution Policy, which allows trademark owners to claim domains that are said to infringe their trademarks, even though the UDRP does not provide all the defences to trademark that ordinary law offers.
The UDRP is pretty much a settled part of ICANN's scope. But there are plenty of other interests (copyright owners, child protection campaigns, law enforcement groups from around the world) that would like ICANN to impose the rules they prefer on domain registrants too. And they're actively lobbying ICANN right now, have been for years.
Under US oversight, there was a principled commitment to the openness of the Internet, and the possibility of an ultimate recourse to Congress if these lobbyists capture ICANN. When that oversight disappears, it will be crucial to have enshrined in ICANN's constitution effective and enforceable means to constrain ICANN from scope creep. Arguments about that are what is delaying the removal of US oversight, with intellectual property lawyers and foreign governments fighting hard to give ICANN a broad Mission that allows it to implement their demands.