In 1964, two researchers at Bell Laboratories, Arno Penzias and Bob Wilson, were observing our Galaxy with a large telescope originally designed to communicate with satellites. They found an additional signal, which was very faint, but was the same from all directions, and didn't seem to change with time. This is odd, since most things either have some specific shape on the sky (like our Solar System or our Galaxy) or they change between seasons, etc.

The Holmdel Telescope. I believe that the two people in the picture are not Penzias and Wilson.

Here is the abstract of their paper:

Measurements of the effective zenith noise temperature of the 20-foot horn-reflector antenna (Crawford, Hogg, and Hunt 1961) at the Crawford Hill Laboratory, Holmdel, New Jersey, at 4080 Mc/s have yielded a value of about 3.5 K higher than expected. This excess temperature is, within the limits of our observations, isotropic, unpolarized, and free from seasonal variations (July, 1964 - April, 1965). A possible explanation for the observed excess noise temperature is the one given by Dicke, Peebles, Roll, and Wilkinson (1965) in a companion letter in this issue.

Dicke, Peebles, Roll and Wilkinson's paper referred to above can be found here. It appeared in the same journal, just above that of Penzias and Wilson. It is in this paper that the "excess" seen by Penzias and Wilson is identified as the Microwave Background.

Penzias and Wilson were awarded the 1978 Nobel Prize in Physics for this discovery.