Joseph,

How can that be?

Through trusts in Delaware, since the owner must be a US citizen.

Does
this mean that this aircraft is under FAA rules and jurisdictions?

Partly. Operation must also be according to local rules (e.g. airpsace and
such), but maintenance or pilot qualification is according to US rules.

What are
the advantages/disadvantages for a foreign owner to do this?

See above. Rules are much more restrictive in many countries than in the US.
Certificates in the US can be much cheaper than in many countries. Thus, it
can be cheaper to operate and fly a US registered aircraft even if you have
to go through the trouble of finding US certified mechanics, inspectors and
repair stations for maintenance.

--
Thomas Borchert (EDDH)