Basically, the particles stood still as the space between them expanded at an exponential rate. Note that this inflation was effectively at more than the speed of light, but since the expansion was on the geometry of the Universe itself, and not the matter, then there is no violation of special relativity. However, those other bubbles are not physically real since they are outside our horizon. We can only relate to them in an imaginary, theoretical sense. They are outside our horizon and we will never be able to communicate with those other bubble universes.
Inflation solves the flatness problem because of the exponential growth.
Imagine a highly crumbled piece of paper. This paper represents the Big Bang universe before inflation. Inflation is like zooming in of some very, very small section of the paper. If we zoom in to a small enough scale the paper will appear flat. Our Universe must be exactly flat for the same reason, it is a very small piece of the larger Big Bang universe.