The Big Bang theory predicts that the early Universe was extremely hot and full of very high energy radiation. As the Universe expanded, space stretched and this radiation was stretched with it. Its wavelength grew longer and longer until it reached the microwave part of the electromagnetic spectrum. So the theory predicts that faint microwave radiation from the early Universe should still be reaching us, coming from every direction in space.
In the 1960s this radiation was detected. It is called the cosmic microwave background (CMB) radiation. It arrives almost equally from all directions and matches a very low temperature of about 2.7 kelvin.
The Steady State theory has no reason to predict the CMB, because in that theory the Universe never had a hot, dense early stage. The Big Bang theory explains red-shift and the CMB, while the Steady State theory cannot explain the CMB. With more evidence supporting it, the Big Bang theory became the currently accepted model for the origin of the Universe.