why do LCDs multiply frame rate?

I believe it's because a 60Hz refresh rate is needed to maintain the 30fps video frame rate for tv's. A 120Hz tv multiplies the 60Hz by two (displays each frame twice) to smooth out motion. Cinema is shot at 24fps but needs to be "sped up" to 30fps for display on tv's. A blu-ray movie can be displayed at 24fps on tv's that are equipped to handle the signal (most, if not all, modern tv's can now). This is, of course, a very simplistic explanation.

Also, 24 fps DOES flicker!

Not on a blu-ray it doesn't.

24 frames per second will appear as a flicker to normal peripheral vision.
In a theater or with a large television, this is most acute because so much of the image is in the periphery.
The brighter the image, the more evident the phenomenon will be, owing to the larger difference in the transition.
The Lord of the Rings was filmed with 48 frames per second to show the improvement.
24 FPS was a standard that barely passed the 'persistence of vision' test and only for the central part of the vision.
After 150 years, there is no reason not to bump this rate...

24fps does not mean that the LCD backlight is flashing at 24fps (that *would* flicker horribly). It just means that the image is changing at 24fps. The backlight is either not flashing at all (absolutely no flicker - which also maximizes brightness) or flashing at 120Hz or higher. A really low frame rate would look like a slide show rather than a movie, but it would not flicker.

Film projectors don't flash at 24Hz either. They flash at either 48Hz or 72Hz, i.e. each frame is illuminated 2 or 3 times.

All LCD TVs have a single fixed frame refresh rate. When receiving 24fps content on a 60HZ TV 1/2 of the frames are displayed 3 times and the other 1/2 are displayed twice since 12x3 + 12x2 =60. On a 120Hx LCD each frame of 24fps content can be displayed 5 times since 5x24=120. This produces 24 different images per second maintaining the 24fps input frame rate.