What does a compressor of a jet engine do?

Just as any air "compressor" does. It compresses air molecules. The same theory can apply the a garage air compressor when it pumps air into a tank. What happens is air is pulled through a convergent duct into the combustion section using a compressor rotor and stator vane cases.

The compressor rotor is a series of discs with small airfoil shaped blades on the end with 2 split half cases containing guide vanes that direct airflow thru the compressor clamped around the outside.

The compressor rotor has a silhouette shape similar to a Christmas tree without the cases installed. That is what helps create a convergent duct that air is compressed thru. The inside shape of the cases 'when assembled' mirror the contour of the rotor.

Basically the compressor squeezes more air molecules into a tighter place to allow more oxygen inside the combustion chamber where the air and fuel is atomized or mixed together so it can be ignited.

The compressor section of a jet engine with "compressor variable vanes" also controls airflow thru the engine. Basically what that means is it keeps the air from going backwards thru the engine by constantly controlling the velocity of air traveling thru the engine.

By changing the position of the "variable vanes" the velocity of air traveling thru the engine changes. The basic theory is by increasing the velocity of the air flow, pressure decreases inside the compressor and by decreasing the velocity of airflow you increase the pressure inside the compressor. Therefore keeping the air moving properly thru the engine.