A gas dissolved in oil is in equilibrium with the same gas in the free space above it. The Ostwald coefficient is the ratio between the concentration in the liquid and the concentration in the gas, at a given temperature.
It matters because every gas has its own and they differ widely: hydrogen partitions in a way that clearly favours the gas phase, while acetylene and ethylene stay mostly dissolved. A reading taken in the gas phase and one taken in the oil do not give the same number even when the transformer is in the same condition.
Hence it is indispensable when comparing results from different methods — vacuum extraction, headspace, online monitors with a membrane — and when converting the reading of a monitor installed in the gas space into concentration in the oil, which is how the diagnostic criteria are written. The coefficient depends on temperature, so the thermal correction is part of the calculation.