To me the best intuition for almost periodicity is Weil’s (1940, Chap. VII), nicely exposed in Dixmier (1982, Chap. 16): Any topological group $\mathrm G$ maps to a “universal” (“Bohr”) compact group $b\mathrm G$, through which all morphisms $\varphi$ of $\mathrm G$ to compact groups (or all finite-dimensional representations of $\mathrm G$) factor uniquely:
Now almost periodic functions on $\mathrm G$ are just those $f$ that factor through a continuous $\tilde f$ on $b\mathrm G$, and $f$’s mean is just $\tilde f$’s Haar integral. (When $\mathrm G$ is locally compact abelian, $b\mathrm G$ is the Pontryagin dual of $\mathrm G$’s dual-made-discrete.)
Weak almost periodicity is a variant for which Glasner (p. 47) gives some references; one could add the “very substantial survey” of Štern (2005), which mentions applications to ergodic theory.