It's all about Freedom:
With typical Closed-Source software, your are beholden to your supplier because they keep the source code (the technical gubbins that descibes the program) secret from you. This gives your supplier power over you, because you have to ask them for fixes, ask them for updates. They might even stop supporting the version of their software that you use, and force you to upgrade to the latest version, and charge you for the the pain they're causing you.
With Open Source Software, you have Freedom.
Because the software supplier gives you the source code of the application, you are not beholden to them. You can use the software for as long as you want, without them forcing you to upgrade.
Sick of your supplier ? Sack the supplier, and get a new one ! Because you have the source code, any technically skilled supplier will be able to support your system.
Techies love Open Source. It lets them have the Freedom to see the details of how the software works, allowing them to tweak the usability, fix bugs, and understand how the application works, which helps if it isn't doing what you expected.
Don't I have to give away my company secrets ?
No - if you use and modify Open Source software, you can keep your company data and modifications secret. If you decide to distribute your changed software, then, "yes", you also have to distribute any changes you made to the original. But most companies don't distribute software, they just use it.
For a Formal Definition of Open Source:Open Source Initiative (OSI) is a non-profit corporation dedicated to managing and promoting the Open Source Definition for the good of the community.