Birgulino and Funster hit on good reasons - frivolous litigation threat is an especially big one. The laws of the foreign country combined with foreign corporations and trusts can serve as a shield. The international law aspects of it get very complicated, but generally a US court must have jurisdiction to enforce a judgment...
I'm speculating, but there may also be some estate planning/inheritance benefits offshore, especially if an individual owns property internationally.
another is financial privacy - which is now essentially nonexistent in the US...