Why do countries not bound by US law comply with the FAA's originate airworthiness directives?
International air travel requires compatible laws at both ends, otherwise a plane might be grounded when it arrived. A web of international agreements therefore exists to implement globally the directives of the major national certificating authorities such as the American FAA and European EASA.
Some planes made in Russia and China are not able to fly internationally to any significant extent because they do not meet FAA or EASA standards. Others have been certified by these authorities and do frequent Western airports.
The agreements are voluntary. For example the Boeing 737 MAX scandal led many national authorities to reject the FAA's judgement to let it keep flying immediately after the second crash, and to make their own judgements when the FAA re-certifies it, while the EASA has been negotiating hard to ensure that the fixes are also compliant with EU standards.