The CMB was first proposed in 1948 by Big Bang supporter George Gamow and his colleagues (although previous flawed estimates of space's temperature had already been calculated). In 1964, a research team at Princeton began building a Dicke radiometer in an attempt to detect the CMB, but they were beaten to it by an accidental finding by Arno Penzias and Robert Woodrow Wilson of Bell Telephone Laboratories. They detected a 3.5 K temperature with their equipment which could not be accounted for, until they learned of the CMB and realized what they'd detected. This discovery essentially ushered in the end of the steady state theory, which would allow for a universe with no beginning and earned Penzias & Wilson the 1978 Nobel Prize in Physics.
Over the years, various projects have re-measured the CMB with greater accuracy, such as the WMAP project which detected minute variations in this temperature throughout the universe.