Blunt answer: a turbine engine does not NEED a compressor in order to work - b ut it does work a great deal better with one.
As mentioned above, it's to do with the efficiency of the combustion process. That applies to combustion engines generally, which is (simplistically) why diesel engines are generally more efficient than petrol engines, why petrol engines using higher grade fuel are more powerful 9because the higher grade allows use of higher compression ratios in the cylinders, hence more power) and why modern gas turbines are more efifcient than the earlier engines, as they run at higher compression ratios.
But there is an example of an engine which doesnt use compression in the combustion process - a steam turbine. You can use atmospheric pressure combustion to create the heat that is used to drive the steam cycle - but it isnt terribly efficient. Yet the turbine will still turn and power is generated.