The "inflationary scenario," developed by Starobinsky and by Guth,
offers a solution to two apparent problems with the Big Bang.  These
problems are known as the flatness-oldness problem and the horizon
problem.

The flatness problem has to do with the fact that density of the
Universe appears to be roughly 10% of the critical density (see
previous question).  This seems rather fortuitous; why is it so close
to the critical density?  We can imagine that the density might be
0.0000001% of the critical value or 100000000% of it.  Why is it so
close to 100%?

The horizon problem relates to the smoothness of the CMB.  The CMB is
exceedingly smooth (if one corrects for the effects caused by the
Earth and Sun's motions).  Two points separated by more than 1 degree
or so have the same temperature to within 0.001%.  However, two points
this far apart today would not have been in causal contact at very
early times in the Universe.  In other words, the distance separating
them was greater than the distance light could travel in the age of
the Universe.  There was no way for two such widely separated points
to communicate and equalize their temperatures.

The inflationary scenario proposes that during a brief period early in
the history of the Universe, the scale size of the Universe expanded
rapidly.  The scale factor of the Universe would have grown
exponentially, a(t) = exp(H(t-t0)), where H is the Hubble parameter,
t0 is the time at the start of inflation, and t is the time at the end
of inflation.  If the inflationary epoch lasts long enough, the
exponential function gets very large.  This makes a(t) very large, and
thus makes the radius of curvature of the Universe very large.  

Inflation, thus, solves the flatness problem rather neatly.  Our
horizon would be only a very small portion of the whole Universe.
Just like a football field on the Earth's surface can appear flat,
even though the Earth itself is certainly curved, the portion of the
Universe we can see might appear flat, even though the Universe as a
whole would not be.

Inflation also proposes a solution for the horizon problem.  If the
rapid expansion occurs for a long enough period of time, two points in
the Universe that were initially quite close together could wind up
very far apart.  Thus, one small region that was at a uniform
temperature could have expanded to become the visible Universe we see
today, with its nearly constant temperature CMB.

The onset of inflation might have been caused by a "phase change."  A
common example of a phase change (that also produces a large increase
in volume) is the change from liquid water to steam.  If one was to
take a heat-resistant, extremely flexible balloon filled with water
and boil the water, the balloon would expand tremendously as the water
changed to steam.  In a similar fashion, astronomers and physicists
have proposed various ways in which the cooling of the Universe could
have led to a sudden, rapid expansion.

It is worth noting that the inflationary scenario is not the same as
the Big Bang.  The Big Bang predicts that the Universe was hotter and
denser in the past; inflation predicts that as a result of the physics
in the expanding Universe, it suddenly underwent a rapid expansion.
Thus, inflation assumes that the Big Bang theory is correct, but the
Big Bang theory does not require inflation.