The key to extradition between countries is that the accusation needs to be for a crime for which an extradition treaty exists. Between the US and NZ, here is a listing (which is typical of other country treaties with the US): https://internationalextraditi... ... I did RTFA, but did not find a link to the NZ court ruling to confirm the extent to which this bilateral extradition treaty was the basis for the ruling.
Dotcom is accused of racketeering and money laundering, which would seem to be covered in the treaty section on fraud: "16. Obtaining property, money or valuable securities by false pretenses or by conspiracy to defraud the public or any person by deceit or falsehood or other fraudulent means, whether such deceit or falsehood or any fraudulent means would or would not amount to a false pretense." The definition of racketeering is something like, "dishonest and fraudulent business dealings."
International extradition treaties are part of why plaintiffs and prosecutors seek such high crimes, in their charges. The article links to the US court filing, if you want to see the full list. Another reason is that, in the US, criminal charges are made at the highest possible level of seriousness, so that there will be a plea bargain for a lower charge, rather than bringing a case all the way to the end. Federal prosecutions in the US very rarely result in Not Guilty or in charges being dismissed (under 5%).
That EU law that got struck down yesterday was part of an industry effort to add copyright infringement to the set of laws that would let enforcement cross national boundaries. For copyright, there is no current international extradition (at least, not with the US -- the EU has been doing its own thing). The Berne Convention, and associated treaties under WIPO, are the applicable international treaties for copyright, and do not make provisions for extradition or international enforcement for copyright violation. The fact that international boundaries are usually very easy to cross via Internet traffic is a big concern for publishers, media companies, etc., and they have been trying for a long time to extend reach of copyright laws beyond national boundaries.
One of the earliest such cases was in 2000, and involved a US copyright law forbidding reverse engineering of encryption. The DeCSS case, https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.technewsworld.com%2F..., was to bring charges against Jon Johansen in Norway for posting a decryption program. Nowadays, I would expect charges in US courts would also include crimes for which extradition treaties apply, like fraud and larceny. This is easily achieved by stipulating large $ damages (due to lost revenue, piracy, etc.).
More recently, we know that Julian Assange is concerned about being extradited to the US under a secret indictment in the US courts. The rape charges in Sweden were sufficient for extradition from the UK (https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.government.se%2Fgovernment-of-sweden%2Fministry-of-justice%2Finternational-judicial-co-operation%2Fextradition-for-criminal-offences%2F), but Ecuador has an approach that gives higher priority to avoiding torture than the bilateral treaties. The Guardian has a nice short cheeky piece about why Edward Snowden was also thought to be en route to Ecuador, before he ended up staying in Russia: https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.theguardian.com%2Fwo...
We are getting the picture, right? The US isn't the only country that seeks very high-level crimes in what are basically copyright cases, nor are they the only country where moneyed business interests are able to get the ear of criminal courts for issues that are, essentially, civil cases (a distinction that matters a lot in countries that follow common law... less so for countries with different legal heritage, like Germany).
In the US, the first thing the big media companies or other large copyright holders do when considering a case is to count up the potential damages. Once they are in the $millions, they can get federal criminal courts and the federal police (the FBI) interested. An early case where this technique was used was the 1995-1999 prosecution and incarceration of Kevin Mitnick (https://ancillary-proxy.atarimworker.io?url=https%3A%2F%2Fwww.wired.com%2F2012%2F02%2Ffeb-15-1995-mitnick-arrested%2F). He had exfiltrated source code and other "trade secrets" from Sun and Motorola, which valued that code in the $100s of millions. It was those incredibly high damage estimates that inspired the US federal prosecutors to pursue a case against Mitnick.