When people talk about outsourcing, they are typically talking about "offshore" outsourcing in which companies from a rich economy send some of their functions overseas. For example, Nike has sent its manufacturing operations overseas and many companies have sent their customer service centers overseas as well.
The main argument in favor of this is that it gives consumers in the rich country the best possible prices. If Nike were to make shoes in the United States, they would cost a great deal more than they do. If companies like Microsoft had to pay American wages to all of their customer service staff, they would have to charge higher prices for their products. Therefore, outsourcing actually helps all of the consumers in a country. They all get to benefit from the lower prices gained by outsourcing. This improves their material quality of life.