This is an example of Newton's 3rd Law: For every reaction there is an equal and opposite reaction. The force of the exhaust gases pushing backward produces an equal and opposite force that propels the plane forward. This force is called thrust.
The jet engine takes in air as it moves through the atmosphere. The air is compressed and mixed with jet fuel. Combustion of the jet fuel/air mixture produces the exhaust gas that leaves through a nozzle at a high speed. The design that increases the speed of the exhaust gas allows it to provide enough thrust to move the much more massive airplane forward. The moving gas also turns turbines that power compressors.
You can observe an example of this phenomenon if you inflate a balloon then let go of it without tying the end. As the air escapes and pushes back the opposite force propels the balloon forward.