Guth's Solution

Microwaves from opposite sides of the sky are the same temperature because they once were in thermal equilibrium.

In 1981, Alan Guth proposed inflation as a solution to several problems associated with the Hot Big Bang model. Inflation allows a small region of the Universe to exponentially increase in size while the Hubble scale remains fixed, thus a volume small enough to establish thermal equilibrium could expand to be much larger than the size of the present visible Universe.

Another way of couching this solution is to say that because of inflation, light could travel a much greater distance between the Big Bang and decoupling than it could between decoupling and the present.