The Berkeley Software Distribution, first released in 1978, was a very important early open-source project. The second version, released in 1979, included the vi editor. When Sun Microsystems formed, they incorporated BSD into their OS, SunOS. In fact, people on the internet have asserted that the TCP/IP stack for most operating systems, including the first stack incorporated into Linux and into Windows 95, was copied from BSD's stack. Since sharing source code is an effective way to increase the chances that two different systems will be able to interoperate on the internet, this TCP/IP stack from BSD (whose creation was funded by the US government) probably materially sped up the growth of the internet.
The original BSD license, which is quite close to the BSD license used by many open-source project today, was created because the owners of BSD (the University of California) believed BSD had to be placed in the public domain because its creation was funded by the US government, and the US government is prohibited by law from copyrighting its creations. Actually, they did not release it into the public domain, but the license they chose is very close to "public domain" in that the restrictions it imposes are very light.
Before 1993, most of the software used for communication on the internet was either explicitly open source (i.e., under the GPL or the BSD license) or "informal open-source", meaning that it did not come with formal license terms, but its source code was available, and it was widely believed that it was unlikely that anyone would try to stop you if you modified it or redistributed it.