Why air is compressed in Gas turbine engine ?

To understand why we compress air in a jet engine, one must first understand the goal. For an aviation gas turbine, the goal is thrust. The purpose of the turbine stage is to power the fan and the compressor. The thrust comes from moving air through the engine.
Now, we know why the fan is there. It's there to move the air. We get thrust from that. Now, the compressor's main purpose is to assist the turbine in energy extraction. By compressing the air, not only does it assist in combustion, it also helps in energy extraction by the turbine stage(s).
As for the temperatures, a turbojet will have significantly higher higher temperatures, as 100% of the flow goes through the turbine stage. As we mentioned, for a thrust-producing aviation engine, the goal of the turbine is to extract 'just' enough energy to power the fan/compressor stages. So, the flow still has a ton of energy, in the form of velocity and temperature. Exhaust temperatures can be well over 1000 degrees.
In a turbofan, as mentioned, some, if not almost all (in the case of the bigger engines) of the flow goes through the bypass. So, while the inner core flow will be hot, after the mixing region, it is relatively mild.