Open source software is software whose source code is both published and freely available to anyone to use, modify, examine, or redistribute as they wish. Such sources are often distributed using an Open Source Initiative (OSI)-approved license, such as the GPL, LGPL, or a variant of the BSD license. Open source software is usually, though not always, free software. Free software licenses grant the user the "four basic freedoms" as defined by the Free Software Foundation, which, while based on a different philosophy, largely overlaps with the open source criteria. Software that meets both philosophies and is licensed to meet both criteria is commonly known as free and open source software (FOSS).
Any programmer can freely modify such sources and release the modifications, with some catches depending on the license chosen by the copyright holder. For example, the GPL, as a copyleft license, demands that modifications are released under the same license, and the Qt license forbids people of creating code branches.
Open source software is differentiated from source-available software, which has published sources allowing users to examine it. However, its licenses, like the one used for Monkey's Audio sources, forbid the user basic modification and/or redistribution rights granted by Open Source and Free Software licenses, and as a result are still proprietary software. Simply having publicly visible source code is not sufficient for software to be considered open source; there must actually be a reasonable ability for the public to use the source in a useful way.
In the context of audio formats, benefits of the use of free and open source software and codecs include user freedom, ease of development/integration, less dependence on the willingness of centralized entities to maintain it, and a greater assurance of the ability to use the codec or software in the long-term as hardware and software advances.
Wikipedia includes more information about open source software and free software.