Usually, the way that the European Union works is to set standards that member nations must follow though its internal political processes (e.g. the European parliament and European commission and specialized areas) that it is a mandate to do under a general E.U. treaty.
Once those standards are set, E.U. member nations are required to adopt domestic legislation or regulations that implement those standards in their own legal system. Then, the laws or regulations mandates by the E.U. are enforced like any other law or regulation of the member country.
Often the domestic legislative of a country applies not only to people in the country that enacts it, but also in any country that does business with the country that passed the law, and generally, someone who does business with a country is subject to the jurisdiction of its laws in its courts.
The failure of someone in a non-European country to comply could lead to a lawsuit against them in Europe and a seizure of their European assets (and the person's assets in any of the many other countries in the world that recognizes the judgments of European courts), and might, in general put that person in a position of complying with European laws and court judgments, or not having any practical ability to do business with Europeans at all.