My primary area of interest is not trade or business history but maritime law, so my knowledge in this area is rather general. I'd recommend this book if you'd like further general reading.

So the most basic question is: what is outsourcing? This is actually a tricky question. Generally speaking we can define outsourcing as shifting operations that were initially done within a firm to another firm. Originally this was done to reduce costs based on (a) scale and (b) specialization, but other factors such as local wages, as you mentioned, have come into play in the past few decades. There are many early examples of this type of outsourcing in the publishing industry in the late 19th century, as bookmakers stopped making paper and began to purchase paper from specialized firms. But I take your question to be about international outsourcing due to cost (commonly called "offshoring"), which is slightly more complicated. Some scholars have noted that the corporate practice began in the 1950s, although it did not really take off until the 1980s.

Note: Outsourcing operations should be contrasted with "vertical integration," which used to be the dominant business model for large corporations. Vertical integration meant that corporations would try to control as much of the product/manufacturing stream as possible (e.g., 7-11 used to own the cows that made the milk sold in its stores).

Okay, so why outsource internationally? The long and short of it is: cost. It became cheaper to manufacture/provide services in other countries, so operations shifted oversees. Sure, labor costs in a country like China in 1990 were substantially lower than in the United States in the same period, but there were other things at play as well, such as technological developments, novel telecommunication abilities, resource abundance, liberalization of trade laws, and increasingly mobile capital.

Let's take the example of the intermodal shipping container, which you're probably used to seeing stacked on container ships, riding on rail cars, or being turned into houses. Those containers were developed in the 1950s, and they vastly improved global shipping practices while reducing costs. Those containers can hold any oddly sized item, like cars or Hello Kitty backpacks, but they can all be stacked together. This made shipping much easier, and led to the development of enormous and efficient container ships that could transport anything with fewer crew and less fuel. Moreover, those containers could be loaded directly on to trains or trucks without having to unpack all of the cargo first. For an analogy, think about buying/storing soda. It's far more efficient to carry 3 12-pack cases of soda and put those cases directly into the fridge than trying wrangle 6 individual 6-packs with those pesky plastic rings. The intermodal shipping container reduced shipping costs, and also killed thousands of longshoring jobs in the US.

But it's not as though wealthy/technologically advanced Western companies just went and found poor folks elsewhere to work for pennies on the dollar, so to speak. The governments of many non-Western countries actively sought foreign investment and "offshoring." China, for example, created "Special Economic Zones" with lower taxes, favorable import/export laws, and more relaxed regulations of foreign business integration. Shenzen, which is right across the border from Hong Kong, is one example. In 1979, when Shenzen was made an SEZ, its population was under 50,000--today, it's 18 million. Not only were the SEZ business incentives enticing to overseas firms, Shenzen's location next to Hong Kong (a power player in global finance which was, at the time, a British colony) and HK's deepwater port provided easy access to both global capital and shipping. Moreover, these new cities could "specialize" in certain sectors of the manufacturing economy, which would concentrate skilled and semi-skilled labor, but would also reduce transportation costs within the supply chain.

As to why it wasn't done earlier? It's hard to say for sure, but transportation, communication, and regulated trade and capital markets are probably the best guesses. For most firms it's a question of efficiency. Some scholars point to the idea that outsourcing allows a firm to focus on "core competencies." At a company like IBM or Apple, for example, they might spend less time worrying about semiconductor manufacturing and more on technological innovation, which will make the firm more efficient, innovative, or whatever.