The idea is to reduce the force on the chain just as you shift to a lower gear. Doing so helps the chain move quickly and smoothly to the larger cog (rear shift) or smaller chainring (front shift).
Modern drivetrains do a good job of shifting under load, but you can give them a hand with this technique. Simply reduce pedaling pressure on the stroke during which the shift takes place, then pour on the coals again.
This is especially helpful for a front shift.
If you're in the big or middle chainring on a climb and bogging down, there's a heavy load on the chain. You need a lower gear, but the chain may be so tight that the derailleur can't pull it over to the small ring.
Or, as in Greg's situation, hard pedaling causes the chain to pull off the large ring with such force that it misses the small ring and dumps onto the bottom bracket.
Coordinate your left hand with your legs. As you move the gear lever, soft pedal for one stroke. Keep the crank turning but with less force. The chain will drop cleanly to the small ring. Then resume mashing.
Ideally, you'll anticipate shifts and make them before putting the chain under a heavy load. But when you goof, this technique will bail you out