RJM,
renard already gave the basic answer and the basic formula.
A given mass of air per second comes in through the intake (at what is basically the airspeed) and gets ejected through the exhaust. So to get thrust (forward force) you have to accelerate that air backwards, i.e., it the jet velocity has to be higher than the airspeed.
(Note that "mass flow in" = "mass flow out" ; the mass of the fuel added is negligeable. It's the energy of the burning fuel that accelerates the air, plus the shape of the exhaust nozzle.)
"...the exhaust air exits compressed..."
Ideally, no. You let it expand in the nozzle to the outside atmospheric pressure, thereby accelerating it.
If the pressure at the end of the nozzle is still higher than the atmospheric pressure, the jet will expand further behind the nozzle, without contributing more thrust, so some of the energy in the jet is wasted.
In practice, unless you have a variable nozzle, the nozzle diameter is optimised for cruise, and you accept there will be some loss in the off-design case.
CJ