Hi Blip,
I'm not a professional pilot, but just got a degree in aero eng, so here's my 2 pence:
Gas turbines all work on pressure ratios. Sitting still on the runway the engine has to 'suck' ambient presure air in to the fan, it then compresses it, burns it, takes some work out and throws the air out of the back. How quickly it comes out will depend what the engine was designed to do, but for your question it comes out at 190kts.
At 300kts at runway level (i know... but it removes changing altitude pressures from confusing the situation) the air entering the engine is at a higher pressure than stationary due to the relative velocity of the aircraft. Assuming the engine is working at the same operating point (to keep component efficiencies the same), when you apply the pressure ratios through the engine cycle you'll find the jet velocity is a lot higher than 300kts.
Hope this explains it and i'm sorry if any of the above patronises you... but don't know what you already know!!