In this all-sky map from the COBE satellite, radiation in the Earth’s direction of motion appears blueshifted and hence hotter, while radiation on the opposite side of the sky is redshifted and colder.
Our Earth is not at rest. The Earth moves around the Sun. The Sun orbits the center of the Milky Way Galaxy. The Milky Way Galaxy orbits in the Local Group of Galaxies. The Local Group falls toward the Virgo Cluster of Galaxies. But these speeds are less than the speed that all of theseobjects together move relative to the cosmic microwave background radiation (CMBR).
The map indicates that the Local Group moves at about 600 kilometers per second relative to this primordial radiation. This high speed was initially unexpected and its magnitude is still unexplained. Why are we moving so fast?