Why do stars twinkle? Why does light from planets not twinkle?
— V. Hariharan, Chennai
Twinkling is closely related to what astronomers call ‘seeing' (atmospheric blurring of an image). Both are caused by the turbulent cells in the upper atmosphere: these are little pockets of air that have different density, temperature, humidity etc. than the surrounding air.
The density contrast causes refraction, and as different cells move in and out of your line of sight, the image of the star (which is point-like) is seen to move around from one second to the next.
This movement is seen as twinkling by the eyes; if you take a photograph over several minutes, as astronomers often do, then the image becomes blurred.
The seeing (this blurring) can be as good as 0.5 second of arc at the best astronomical sites on Earth, while the worst seen by two astronomers from NASA at a professional observatory was about 8 seconds of arc (they gave up observing that night; more typically, 2 seconds of arc would be considered bad by today's professional astronomers). 1 second of arc is 1/3600th of a degree.
So why don't planets twinkle? This is because, even though they may look point-like to naked eyes, they are actually much bigger than the typical seeing. This means that you observe the combination of light which has passed through different atmospheric cells. Thus, the turbulent effects are averaged out, making the planets look steady.