The concept relies on peer review to find and eliminate bugs in the program code, a process which commercially developed and packaged programs do not utilize. As a side effect open source can (also) provide superior security than closed source. Compared to proprietary solutions everybody can audit the code and verify the claims made by the vendor about whether the software really does what it intends. With proprietary solutions this isn’t possible.
Probably the best example for open source is the Internet’s infrastructure. Consider DNS, sendmail, apache, mysql, the various open-source TCP/IP stacks and utility suites, and the open-source scripting languages such as Perl, Ruby or Python that are behind most dynamic content on the Web. These are the running gears of the Internet!