What exactly is the problem that Inflation solves? There's copious documents about how Inflation solves the problem that General Relativity predicts a lumpy CMB.  That influation 'smooths' out the curvature fluctuations and, so, predicts a CMB at thermal equilibrium.  The perfect Black Body radiation curve is proof that the visible sky was in causal contact with all other parts of the visible sky at some point.
But I don't understand where this problem comes from and can't find a lucid description in any of the text.  At t=0, the universe was in causal contact and in thermal equilibrium.  It must be a perfect Black Body at this point.  At $t=\delta$, the peak temperature of space is at some finite value, and again, the radiation would give a perfect profile because there's no physics to randomly distribute the initial energy (no prime mover) of the big bang.  This goes on for $t=2\delta$, $t=3\delta$, and so on.
Time passes and we eventually get a quantum fluctuations in the fields.  One of the fields that can fluctuate is energy/mass.  So we get a blip in the curvature because one small section has more/less mass than the average.  Are the inhomogeneities that Inflation fixes caused by the fact that the scale factor is now permanently different in the more/less dense regions?
 A: A black body does not have a uniform temperature. Any object above absolute zero experiences random statistical fluctuations in its energy distribution so it has hotter regions and colder regions. In a body that is in thermal equilibrium these fluctuations are continually being created and disappearing so the temperature overall has a constant and well defined value.
The problem with the universe is that in the early stages of its expansion it was expanding too fast for the hotter fluctuations to heat the colder fluctuations and disappear. As soon as a fluctuation appeared the expansion would tear apart the hotter and colder regions and preserve the temperature difference.
In fact the subsequent expansion increases the inhomogeneity because the colder denser regions expand less and the hotter less dense regions expand more. The end result is that by the time of the CMB the thermal fluctuations would have created temperature differences many orders of magnitude larger than are observed in the CMB. This is the fundamental problem that inflation solves.
A: The problem, or claim is that inflation solves the flatness problem. Concretely, the problem is that in cosmology there’s a dynamical variable (ie, one that depends on time), called the curvature density parameter. It’s by construction dimensionless (doesn’t have units) and its value today is smaller than 0.1.
The standard model of hot big-bang cosmology requires initial conditions which are problematic in two ways: 


*

*The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causally disconnected (horizon problem); and 

*Tthe initial value of the Hubble constant must be fine tuned to extraordinary accuracy to produce a universe as flat (i.e., near critical mass density) as the one we see today (this is the flatness problem!). 


These problems would disappear if, in its early history, the universe super-cooled to temperatures of orders of magnitude below the critical temperature for some phase transition. 
A huge expansion factor would then result from a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when the latent heat is released. Such a scenario is completely natural in the context of grand unified models of elementary-particle interactions.
For further reading see
Lecture 1 here.
