It happened in less than a trillionth of a trillionth of a second. This brief yet dramatic expansion stretched space itself, making the universe uniform on large scales. Quantum fluctuations during inflation became the blueprint for cosmic structures. Inflation solves puzzles like the horizon problem and flatness problem. Without it, the cosmos would look very different today.
It matters because inflation explains the smoothness and large-scale uniformity of the universe.
It also predicts subtle fluctuations we can measure in the cosmic microwave background.
Space itself grew faster than light, even though nothing can locally travel faster than light.
CERN [cern.ch]