Open Source, as we're defining it, is Free software–not free as in free of cost, but free as in freedoms associated with the software.
Non-Open Source programs are generally licensed in an unmodifiable form. If a bug is found or an important feature needs adding, the only way is to work with the vendor who wrote the program. If the vendor is out of business, no longer supports that program, provides poor service, or demands an outrageous rate, you're out of luck.
Open Source licenses are different. With an Open Source license you have the freedom to modify your software. All Open Source projects come complete with the 'source code' for the project. With the source code you can do anything with a program.
While you may pay for an open source solution, you aren't locked into a relationship with a particular vendor. The software itself is always freely available in an open medium that allows for others to further develop it and benefit from it.
For more information about free and open source software and licenses:
