You can think about it as similar to a socket and ratchet. Hold the ratchet in one hand and the socket in the other hand, the ratchet represents the driveline and the socket represents the wheel. If you turn the ratchet the socket has to turn in that direction, but if you turn the socket in the same direction it will click and spin ahead. Exactly the same thing happens in an automatic locker (Aussie, Lockright, Detroit, etc) on each side. The center pin in the diff puts pressure against the side gears of the locker putting outward pressure that forces the dog teeth to stay engaged and the locker locks both axles together. Under light to moderate throttle as long as the wheel trying to spin ahead puts more force back against the side gear than the center pin puts against the side gear then that one side will unlock and that side wheel will spin ahead freely while cornering. If lots of throttle is applied it will lock up.
The unique driving characteristics of an auto locker are because only the inside wheel is getting power around corners, which can feel like a push around corners sometimes or a wiggle in the rear when changing cornering directions on a winding road since the driving wheel keeps changing sides.
Sometimes there is a tire chirp when starting out around a corner, like from a stop sign and making a right hand turn. The tire chirp is because only that inside tire is getting power, the weight is transferring to the outside removing weight from that tire, and a little too much throttle is applied casing that tire to spin until it catches up with the unlocked outside tire and the dog teeth side gears engage again, then the outside tire spins ahead again, the inside tire chirps to catch up, etc, etc, and you ge that chirp chirp of tire spin. If you can be a touch easier on the throttle it won't chirp, or if you really mash the throttle then the locker will stay locked and you'll just get tire spin.