The reason we can get away with throwing away low frequencies in JPEG is because humans are prone to notice significant details rather than tiny details.
High frequencies of a Fourier transform of an image == tiny detail (like being able to distinguish individual hairs)
Low frequencies of a Fourier transform of an image == huge details (like someone's face).
So you transform, set part of the result to zeroes, and compress. To display it you uncompress, transform back, and display it. The zeroes manifest themselves as an almost-imperceptible blur.
Open that image in a new tab, then hold down CTRL and scroll up/down to zoom in and out. When you zoom out to 25%, each of them switches from smiling to frowning or vice versa. The lady on the right is clearly smiling at 25% zoom. I had no idea that was possible.