A company might consider software outsourcing a website's testing for several
reasons... 
1. The company may be launching a new site, or a new version of their site, and may feel that most testing tasks can be relegated to offshore provider.
2. The company may not have the resources -- people, skills, software, hardware or time - to perform testing.
3. The project to be tested may be of such as short life span that the company doesn't need any long-term investment in testing processes.
4. The company might want an independent third party to perform the testing, in order to get a more objective view of site quality.
5. The company may even be outsourcing the development and coding for the site, making the outsourcing of the testing a reasonable decision. (Even more reasonable would be a firm that provides the coding and the quality control for its own code.)
The decision to outsource testing needs to be a well-considered decision, because most people who aren't responsible for the testing misunderstand the meanings and scopes of the concepts involved in testing. Two very important issues must be resolved before taking the outsource step: first, the company that owns the website must be absolutely clear about the scope of the job -- including the tasks, processes, and responsibilities -- that they want to hire out; and second, the company must be sure that they are speaking the same "language" as the test firm they will hire.