I skimmed through the SIFT paper. I understand that there are multiple octaves, which are composed of multiple layers. The layer $k$ of an octave (btw, where does this name come from?) corresponds to the image filtered with a Gaussian kernel with a standard deviation of $k\sigma$, so layers higher in the octave correspond to more blurred versions of the original image. From these blurred images, you calculate the difference of Gaussians (DoG), from which you can compute the pixels that are locally higher (compare to a neighborhood around that pixel). The pixels that are locally higher than its neighbours are the potential keypoints. Once this is done, you can downsample the Gaussian-filtered images in the first octave to produce another octave, then you compute the DoGs again. This process can be repeated (for several iterations).
Intuitively, we apply the Gaussian kernel with a progressively higher standard deviation so that we can find keypoints at different scales (or resolutions), but why do we need to further downsample the Gaussian-filtered images in the first octave and repeat the process? Intuitively, this could still help to find keypoints at different resolutions (or scales), but I can't fully understand why would this approach be correct. So, why do we need multiple layers per octave and multiple octaves in SIFT?