Why do we need multiple layers in each octave and multiple octaves in SIFT?
In music theory, an octave is an interval in frequency, from a frequency $f$ to frequency $2f$. For example &quot;an octave higher&quot; means &quot;twice the frequency&quot;. Expressed as wavelength inversely proportional to frequency, $\lambda \propto \frac{1}{f}$, an octave would be the interval from a $\lambda$ to $\frac{1}{2}\lambda$. In the SIFT paper's notation, $\sigma$ is the scale, which is analogous to wavelength as it is proportional to the width of the Gaussian kernel.
Downsampling is just to reduce the image size so that there is less computation to do. From the paper (emphasis mine):

Once a complete octave has been processed, we resample the Gaussian
image that has twice the initial value of $σ$ (it will be 2 images from
the top of the stack) by taking every second pixel in each row and
column. The accuracy of sampling relative to $σ$ is no different than
for the start of the previous octave, while computation is greatly
reduced.

What enables to do this? A Gaussian function has another Gaussian function as its Fourier transform, and that decays faster than can be counteracted by any differential operator such as the Laplace operator. A finite-amplitude low-pass signal filtered by a Gaussian or a Laplacian of Gaussian or its approximation of scale $\sigma$ has in an approximate sense a band limit that is proportional to $\frac{1}{\sigma}$. By increasing the constant of proportionality, the mean square power of the part of the signal that is above the band limit can be made arbitrarily small. This allows to resample the filtered signal with a sampling frequency proportional to $\frac{1}{\sigma}$ while keeping the aliasing error arbitrarily small. A convenient approach is to not sample every signal filtered at scale $\sigma$ by sampling frequency $\frac{\sigma_0}{\sigma}$ with $\sigma_0$ some reference scale that is sampled at sampling frequency $1$, but instead by sampling frequency $2^{\operatorname{ceiling}\left(\log_2(\frac{\sigma_0}{\sigma})\right)}$:
Figure 1. Sampling frequency (red) is the same for all frequencies within an octave. The drawback is that for some scales the sampling frequency is higher than the actual requirement (blue) for that scale. $\sigma$ is the scale and $\sigma_0$ is a reference scale.
This bunches together values of $\sigma$ into a number of octaves such that within each octave the sampling frequency is a power of 2, ensuring that the location of each spatial sample conveniently coincides with a spatial sample one octave higher. The drawback is that the sampling frequency for all scales in an an octave must be chosen based on the lowest scale included in the octave, to keep the aliasing noise within acceptable limits for that scale. The higher scales in the octave will typically have less aliasing.