Layback slides and slashes are an extreme maneuver that originated with surfing. When surfing, a surfer would literally lay back on a wave and ride it before rising to stand again. As you can imagine it is very hard for a person to lean back against a wall of water and thus the desire to try the trick and adapt to other types riding.
There are two types of layback maneuvers commonly seen, and booth is basically the same. When surfing, a layback slash means lying back against the wave and then slashing the board across the crest of the wave to send water spraying upwards while rising again. The slide is a slightly less extreme version of this where you carve the inside of the wave.
Since this is basically a surfing trick it begs the question, how was it adapted for skateboarding and snowboarding? The first part was to change the surface that was laid back against. Instead of leaning back against a wave, the rider leans back against whatever surface they are riding on and hold themselves up off that surface with their hand. Obviously a good glove is needed for this.
Begin by riding down a slope that you are comfortable with at a highly high rate of speed and begin to carve the hill. As you enter the carve, lean down and place your front facing hand on the hill behind you. As you brace yourself and feel the new balance points, extend your arm even further behind you thus causing your body to stretch out behind you.
After a few seconds begin to retract your arm and raise yourself up off the ground again. Depending upon your body and riding style you may wish to raise by using a crab like style where you arch your back a bit and use your legs or you may wish to retract your arm, crouch down while doing so, and upright yourself. Generally it is easiest to crouch down and then upright yourself.
Once you have mastered layback slides you can begin to work on slashes. Obviously this requires a lip of some type such as a bowl or chute. The beginning of the trick is the exact same as before but you will need to worry more about positioning. You will want to start your carve at a point where the edge of your snowboard is almost cresting the tip of the chute or bowl.
As you carve you will naturally go into a Fakie position and your snowboard will be driven towards the crest of the chute or bowl. As this happens lower yourself and extend your arm behind you as before. Use your arm as a pivot point and raise the edge of your snowboard above the lip of the chute or ramp. When done force your snowboard to point back into the chute or bowl and raise up as you normally would.
As you can imagine layback maneuvers are fairly hard to achieve but are also very glamour and very impressive when completed. Having an extremely advanced snowboarding ability is an absolute must before attempting as is having the proper safety gear. However, the work and effort are more than worth it once you too can do this and impressive everyone on the hill.