The physics of flight make such oscillations more probable for pilots than for automobile drivers. An attempt to cause the aircraft to climb, say by applying up elevator will also result in a reduction in airspeed.
Another factor is the response rate of flight instruments in comparison to the response rate of the aircraft itself. An increase in power will not result in an immediate increase in airspeed. An increase in climb rate will not show up immediately on the vertical speed indicator.
A pilot aiming for a 500 foot per minute descent, for example, may find himself descending too rapidly. He begins to apply up elevator until the vertical speed indicator shows 500 feet per minute. However, because the vertical speed indicator lags the actual vertical speed, he is actually descending at much less than 500 feet per minute. He then begins applying down elevator until the vertical speed indicator reads 500 feet per minute, starting the cycle over. It's harder than it might seem to stabilize the vertical speed because the airspeed also constantly changes.
The most dangerous pilot-induced oscillations can occur during landing. A bit too much up elevator during the flare can result in the plane getting dangerously slow and threatening to stall. A natural reaction to this is to push the nose down harder than one pulled it up, but then the pilot finds himself staring at the ground. An even larger amount of up elevator starts the cycle over again.