Here is one source of intuition for the 8-fold and 2-fold Bott periodicities, from the physics of topological states of matter. See <A HREF="https://topocondmat.org/w8_general/classification.html">this online lecture.</A>

In physics, Bott periodicity governs the classification of the fundamental symmetries of free electrons: the anti-unitary symmetries ${\cal T}_\pm$ (time-reversal symmetry) and ${\cal P}_\pm$ (particle-hole symmetry), and the unitary symmetry ${\cal C}={\cal PT}$ (chiral symmetry). The subscript $\pm$ distinguishes whether the anti-unitary symmetry squares to $+1$ or $-1$.

There are 8 symmetry classes with at least one anti-unitary symmetry: $${\cal P}_+;\;\;{\cal P}_-;\;\;{\cal T}_+;\;\;{\cal T}_-;$$
$${\cal P}_+,{\cal T}_+;\;\;{\cal P}_+,{\cal T}_-;\;\;{\cal P}_-,{\cal T}_+;\;\;{\cal P}_-,{\cal T}_-;$$

There are 2 symmetry classes with no anti-unitary symmetries, one with and one without chiral symmetry:
$${\cal C};\;\;\times;$$

A gapped system on a $d$-dimensional torus can be raised to dimension $d+1$ by adding one momentum component, in such a way that the gap does not close on the torus. This operation transforms the symmetry class in a way that reflects the Bott periodicity, as indicated in the diagram below (the "Bott clock").

<IMG SRC="https://i.sstatic.net/byiEBZUr.png" WIDTH="400"/>