Why do pinball machines/games have a nudge/shake/bump feature?

In real pinball machines, players would bump and tilt the table to tweak the trajectory of the ball to their liking. This is known as "nudging". While a little nudging can add skill and depth, too much breaks the game and needs to be guarded against. This is dealt with using sensors that kill the current ball if the table is tilted or smacked too much. Pinball video games seem to have a nudge feature just to emulate this aspect of real-life play.

Source: http://en.wikipedia.org/wiki/Pinball#Nudging

It's for if the ball gets stuck, i.e. perfectly balanced behind a bumper or wedged behind something. The real equivalent would be physically hitting the table.

In real pinball, once you've played the game a significant number of times, you'll start to identify certain areas in the game that are trouble spots where you are likely to lose a ball. Usually these are oriented around the two outlanes on the side, particularly whether a given ball is going to end up going down the outlane and not the inlane. Each game has slightly different dividers between the lanes, different bumper angles and constructions in the vicinity of the lanes. But regardless of the specifics, applying a slight nudge at just the right time can help influence the ball into the inlane instead of the outlane. Some games also have certain ramp shots that will drain unless they're nudged on the way out of the ramp, especially for a shot that doesn't have enough force and doesn't clear a ramp.

In some games a little bit of nudge is useful for keeping the ball dancing between jet bumpers. But most players only take the risk of a tilt on heavy nudging when they're trying to save a ball.