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On theoretical grounds, astronomers think that the very early universe experienced a time of ultra-fast expansion (called inflation). The inflation took place at about 10-35 seconds after the Big Bang. Before that time, the fundamental forces of the strong nuclear force, the weak nuclear force, and electromagnetic force behaved in the same way under the extreme temperatures. They were part of the same fundamental unified force. Theories that describe the conditions when the forces were unified are called Grand Unified Theories (GUTs for short). At about 10-35 seconds after the Big Bang, the universe had cooled down to ``only'' 1027 K and the strong nuclear force broke away from the weak nuclear and electromagnetic forces. This breaking apart of the forces from each other, produced the huge expansion that expanded the universe by about 1050 times in about 10-32 seconds.
The inflation theory predicts that the ultra-fast inflation would have expanded away any large-scale curvature of the part of the universe we can detect. It is analogous to taking a small globe and expanding it to the size of the Earth. The globe is still curved but the local piece you would see would appear to be fairly flat. The small universe inflated by a large amount and the part of the universe you can observe appears to be nearly flat. That solves the flatness problem. The horizon problem is solved by inflation because regions that appear to be isolated from each other were in contact with each other before the inflation period. They came into equilibrium before inflation expanded them far away from each other. Another bonus is that the GUTs that predict inflation also predict an asymmetry between matter and antimatter, so that there should be an excess of matter over antimatter.