**Q:** *How should one think intuitively about an amenable action?*   
A group or group action is amenable if it has an invariant mean. Measurable (from the original name in German) would have been a more intuitively obvious name.    
<sub>Apparently, the term "amenable" is a play of words on "mean" (in British English *amenable* is pronounced "a-mean-able", hinting at "able to support a mean"), see this <A HREF="https://math.stackexchange.com/a/1307539/87355">MSE posting</A>.</sub>

With reference to the Brown-Ozawa book cited in the OP, the invariant mean for an amenable action is constructed in definition 4.3.5. See also definition 1.2 of <A HREF="https://cel.archives-ouvertes.fr/cel-00360390/document">Anantharaman-Delaroche</A> for this construction.