Confusion over off-shore banking begins with not understanding the monetary system of one’s own country. Other than small quantities of cash, US dollars only exist within the Federal Reserve banking system and get shuffled around between various account holders within that system. Many countries have accounts in our banking system because they get paid in dollars via international trade and can use those dollars to pay others who also have accounts in our system. However, those nations cannot move those dollars into their own currencies to transfer them into their own banking systems without paying large fees and dealing with brokers who depend upon shifting valuations for their profits.
Many American companies do business in other countries and many foreign companies do business in America. When taxing profits made by those companies most countries allow the exclusion of profits made in other countries when the product is not sourced or sold in their nations. While this encourages international trade it also offers opportunities to shift costs and profits between the subsidiaries of the companies to reduce tax exposure. Figures never lie, but liars are usually very good with figures.