What is the meaning of the business term "offshore outsourcing?"

The thing to be careful of, here, is in the definition of "offshore."  We tend to think of "offshore" as meaning somewhere out in the ocean or something, like on a ship or oil rig.  In the sense of "offshore outsourcing," the term means "someplace other than a company's home country."  If it is being related to a company in the US, than "offshore" would refer to any place other than the US.

You really have two similar but distinct business terms here: outsourcing, and offshoring. Outsourcing is the practice of hiring another firm to run some aspect of your business, such as the accounting or customer service aspects.  Offshoring is the practice of transferring some aspect of you business to a foreign country, where it may or may not be handled by the company doing the offshoring.  Confused?

Here's an example of Outsourcing: Company A, a small lightning rod manufacturer, decides that it is not cost effective to maintain a bank of phones for customer service purposes.  Apparently, few people call to complain.  Because it is cheaper than paying a bunch of people to sit all day in front of silent phones, they contract with another company (we'll call it "Complain Co") to handle customer complaints for them. "Complain Co" also represents 50 other small businesses that aren't big enough to need their own complaint numbers.

Here's an example of Offshoring: Company A lightning rods grows really big and needs a big complaint center again.  Instead of paying American workers $10 an hour to answer phones, it either opens an office in India and pays workers $2 an hour to answer phones, or hires a company already in India to do the same thing.  Dealer's choice.

Does that make sense?   There are lots of reasons why companies do this, but the two biggest seem to be that it saves money and allows smaller companies to focus on doing the type of business they are good at, rather than the worrying about some aspect of the business that they don't have as much experience with.  While this practice reduces the cost of doing business for a company, it also eliminates American jobs, just leading to a nasty cycle by which there is less money in the economy to be spent (which leads to more outsourcing/offshoring in order to squeeze out profit.)

In the end, blessing or curse, outsourcing is probably here to stay.