The Cosmic Microwave Background (CMB) radiation is the afterglow of the Big Bang. It was emitted when the universe was approximately 380,000 years old. The reason we can observe the CMB today, even though it was emitted billions of years ago, is due to the expansion of the universe.
After the initial emission of the CMB, the universe continued to expand, causing the wavelengths of the radiation to stretch or "redshift" as space itself expands. This expansion effectively stretches the wavelengths of the CMB photons, shifting them from higher energy (shorter wavelengths) to lower energy (longer wavelengths).
As a result of this redshifting, the CMB photons have been stretched over time to longer wavelengths in the microwave range, hence the name "Cosmic Microwave Background." Today, the CMB radiation has cooled to a temperature of approximately 2.7 Kelvin, which corresponds to microwave wavelengths.
Since the CMB photons have been traveling through space for billions of years, they have had ample time to reach us. However, it is essential to note that the universe has also expanded during this time, which has caused the photons to stretch further and decrease in energy.
Therefore, while the CMB was emitted approximately 380,000 years after the Big Bang, the expansion of the universe has stretched its wavelength, and it has taken billions of years for the redshifted CMB photons to reach us on Earth. This is why we observe the CMB as a pervasive background radiation throughout the universe, originating from the early stages of its existence.