Why is CMB an evidence of hotter and denser early Universe?
Beyond the fact that the cosmic microwave background (CMB) is a direct prediction of the big bang model, there is the question of how you would produce it in any other way. It is remarkably close to being isotropic and remarkably close to being a blackbody spectrum - i.e. it is almost a perfect blackbody radiation field.
A blackbody radiation field is emitted by material in complete thermodynamic equilibrium (CTE). An example would be the interior of a star. A requirement for (CTE) is that the matter and radiation field are characterised by the same temperature and that the material is "optically thick" - meaning that it is opaque to that radiation at basically all wavelengths.
Given that the universe is mainly made up of hydrogen, helium and (presently) traces of heavier elements, we can ask how is it possible to produce a perfect blackbody radiation field? Cold hydrogen and helium are transparent to microwaves. To make them opaque they need to be ionised, so that the free electrons can be a source of opacity at all wavelengths via Thomson scattering. But this requires much higher temperatures - about 3000 K.
How do we uniformly raise the temperature of a gas (adiabatically)? By squeezing it. A smaller, denser universe would be hot enough to have ionised hydrogen and would be opaque to the radiation within it. As it expanded and cooled, the electrons combined with protons to form atoms and the universe becomes transparent, but filled with a perfect blackbody radiation spectrum. The light, originally at a temperature of 3000 K and mainly in the visible and infrared, has had its wavelengths stretched by a factor of 1100 by expansion of the universe, meaning we now see it mainly as microwaves.
Additional evidence for this model is that the radiation field is not absolutely isotropic. These small ripples encode information such as the expansion rate of the universe at the time of (re)combination and the density of matter. When inferred from measurements, these parameters agree very closely with other determinations that are independent of the CMB, such as the Hubble redshift distance relationship and estimates of the primordial abundance of Deuterium and Helium.
There is now direct evidence that the CMB was hotter in the past and by exactly the amount predicted by an adiabatic expansion. The source of this evidence is measurements of the Sunyaev-Zel'dovich effect towards galaxy clusters (e.g. Luzzi et al. 2009); or more precisely by probing the excitation conditions in gas clouds at high redshift using even more distant quasars as probes (e.g. Srianand et al. 2008.
The Big Bang model of an expanding Friedmann universe, with a hot and dense early era that cooled and thinned as the universe expanded, predicted the existence, isotropy, and approximate temperature of the cosmic microwave background. So the discovery of this CMB is considered strong confirmatory evidence for that remarkably simple and elegant model. See this list of predictions of the CMB temperature.
This paper reviews alternate explanations of the CMB, and why they are not considered as persuasive as the Big Bang model.