One of the experimental evidence that supports the theory of big bang is cosmic microwave background radiation (CMBR). From what I've read is that CMBR is the left over radiation from an early stage of the universe.
My questions are:
Why are we able to detect this radiation at all?
As the nothing travels faster light, shouldn't this radiation have passed over earth long time ago?
Why is this radiation filling the universe uniformly?
This radiation was created 380,000 years after the Big Bang at every place of the Universe and from every place of the Universe, it was moving in every possible direction. So the density (per unit volume and per unit solid angle of motion) of photons at a particular place $(x,y,z)$ and a particular direction of motion $(k_\theta,k_\phi)$ was always constant: $$ \rho(x,y,z,k_\theta,k_\phi) = {\rm const} $$ By translational and rotational symmetry, it follows that the evolution of this density of photons stays constant as a function of position and direction at all times i.e. $$ \rho(x,y,z,k_\theta,k_\phi;t) = f(t) $$ It only depends on time. The photons we see right now are photons that were created 380,000 years after the Big Bang – a universal moment. They were created in the direction from which they're coming. But another question is how far is the point where the photons we observe were created. They were created at a big distance from us – exactly the right distance from the Earth so that after 13.7 billion years, they manage to hit our satellites.
As the Universe is getting older, we are observing CMB photons that were created at an increasing distance from the Earth. Note that $\rho$ above also depends on $\omega\sim |\vec k|$, the frequency of the photons; this dependence is given by the black body curve while the temperature is dropping inversely proportionally to the linear distances between things in the Universe that keep on increasing.
@Lubos is correct, but here is some additional information about why all these photons started traveling towards us when the universe was only 380,000 years old. Up to that time the temperature of the universe was over 3,000 $K^o$ (approximately - this is from my memory which may be slightly inaccurate) and all the hydrogen and helium atoms were ionized. So up to then the Univese was opaque to radiation and the radiation and the ionized plasma that filled the universe were in thermal equilibrium.
When the temperature dropped below 3,000 $K^o$ the electrons and nuclei recombined to form neutral atoms. At that point the neutral atoms became transparent to the radiation so the photons of that radiation which had the spectrum of a blackbody at 3,000 $K^o$ started streaming freely through the universe till they hit our detectors that are measuring the CMB today.
In the meantime the expansion of the Univese has redshifted those photons from a blackbody at 3,000 $K^o$ to the temperature of 2.7 $K^o$ that we observe today.
