While computers were not a new concept in the 1950s, there were relatively few computers in existence and the field of computer science was still in its infancy. Most of the advances in technology at the time - cryptography, radar, battlefield communications - were due to military operations during World War II, and it was, in fact, government activities that led to the development of the Internet.
On October 4, 1957, the Soviets launched Sputnik, man's first foray into outer space, and the U.S. government under President Eisenhower subsequently launched an aggressive military campaign to compete with and surpass the Soviet activities. From the launch of Sputnik and the U.S.S.R. testing its first intercontinental ballistic missile, the Advanced Research Projects Agency (ARPA) was born. ARPA was the U.S. government's research agency for all space and strategic missile research. In 1958, NASA was formed, and the activities of ARPA moved away from aeronautics and focused mainly on computer science and information processing. One of ARPA's goals was to connect mainframe computers at different universities around the country so that they would be able to communicate using a common language and a common protocol. Thus the ARPAnet -- the world's first multiple-site computer network -- was created in 1969.
The original ARPAnet eventually grew into the Internet. The Internet was based on the concept that there would be multiple independent networks that began with the ARPAnet as the pioneering packet-switching network but would soon include packet satellite networks and ground-based packet radio networks.
For further information on the origin of the Internet, see:
A Brief History of the Internet
A Brief Timeline of the Internet
Short History of the Internet by Bruce Sterling