DCTs are very useful at energy compaction, so simply put after a DCT of an image is resolved to a weighted some of some basis functions. After a DCT, the resulting matrix will contains multipliers for each basis function. And one can without loss of generality say that the high value coefficients are the ones that contribute significantly to the psycho-visual perception of the image by the human eye.
Low frequency noise will add to the low frequency coefficients, however high frequency noise will result in smaller magnitudes of the resulting transformed matrix's high frequency coefficients.
So when we magnitude threshold the transformed matrix, we eliminate all noise that isn't a part of the high magnitude coefficients. So some noise will still be present that may be apparent after the IDCT.
But the main idea here is in images where high frequency data is minimal, a DCT, followed by magnitude thresholding will probably do better than a typical High Pass Filter. If one can imagine an image where any frequency in the image has a real image component and a noise component, where the real image component is small or zero, a DCT followed by magnitude thresholding will eliminate that frequency component, thereby mostly targeting the noisy component.