Doing business with US customers in US dollars and interacting with US banks automatically places you under the jurisdiction of US laws.
Investigators in EU countries often work with US agencies because our laws are so much stricter, and often urge people under investigation to cooperate otherwise they'll turn over the case to the Feds.
Tom Hayes, of LIBOR scandal fame, spilled his guts and signed a confession after being told he might be handed over to the Americans. Then he tried the aspergers defense, followed by the "I only confessed because they threatened me" defense, but neither swayed the jurors and he got 14 (reduced to 11) years in a UK prison.
That's not unique to the US. Countries own their currency and regulate their banks therefore using them gives a country the right to control what you do.
This is grossly oversimplified but generally speaking, if I lived in France and started a payment service that accessed the Japanese banking system and performed transactions in Yen, I would be subject to Japanese banking laws as well as French laws.
[1]: https://en.wikipedia.org/wiki/MindGeek