The air is compressed in the compressor section of the engine and heats up. Hot air is able to hold disproportionally more water in gaseous form than cold air, so all that water will vaporize and enter the combustion process together with the air. Since it is already vaporized, it will not absorb that evaporation enthalpy again that makes water such a good extinguishing agent. On the other hand, since the evaporation enthalpy was sucked from the air in the compression process, the air will be cooler when it enters the combustion chamber, so more fuel can be burnt (up to the stoichiometric ratio of course) and power can be increased.
This trick was used on high performance piston engines and early jets. The Boeing 727 would inject water into the compressor on purpose on hot and high take-offs to increase engine thrust.
If the amount of water entering the engine overwhelms the absorbing capacity of the hot air, the engine runs the risk of flaming out. Regular precipitation does not pose this threat, however.