The LED has a fair bit of junction capacitance, probably 50-100 pF, and at subthreshold voltages it will appear capacitive in both directions, preventing any DC voltage from building up (unless/until you hit the diode with a lot more power than it receives here.)
By shorting out the negative half-cycles (neglecting Vf), the Schottky forces the LED junction capacitance to charge only on the positive half-cycles, allowing enough voltage to build up to turn the LED on.