It is impossible to understand the motivation behind the definition of an amenable action without first understanding the definition of amenable groups, so let me first talk about groups (for simplicity, just countable ones).

Finite groups are precisely the ones for which there is an invariant probability measure (the uniform one). Infinite groups also have invariant measures (the counting ones), but they can not be normalized and made finite (for topological groups one should talk about the Haar measures instead of the counting ones).

There are two natural ways of relaxing the above condition of the existence of a finite invariant measure, and therefore of extending the class of finite groups. One consists in extending the space of measures and looking for invariant objects in this extended space. In our context it amounts to passing from the "usual" sigma-additive probability measures to finitely additive ones (called means). This is precisely the original 1929 definition of von Neumann: a group is amenable if it has an invariant mean.

The other way consists in keeping measures, but replacing, as a trade off, exact invariance with an approximate one. In spite of being more constructive, this approach was only developed in the 50s, and turned out to be equivalent to the original one (I skip the historical details and just mention that the most important contributor was Day who also coined the modern term "amenability"). More precisely, a group $G$ is amenable if and only if it carries an approximately invariant sequence of probability measures $m_n$, i.e., such that $$ \| g m_n - m_n \| \to 0 \qquad\forall\,g\in G \;. $$ This is the Day condition (sometimes one also adds the name of Reiter, but this is not really correct from the historical point of view).

In order to pass to group actions, it is actually more convenient to go a bit further and first talk about groupoids instead. They are like groups with the only difference that the multiplication is defined only subject to a certain condition. Very succinctly, a groupoid is a "small category with invertible morphisms". In plain language it means that there is a set of points ("objects") and a set of arrows ("morphisms") between objects endowed with a composition operation with the same properties as for groups and with the only additional requirement that for composing two arrows the endpoint of the first must coincide with the starting point of the second. In particular, a part of the formal definition of a groupoid is the presence of a target map from morphisms to objects which assigns to any arrow its endpoint. Therefore, over each object $x$ of a groupoid $\mathbf G$ there is the corresponding fiber $\mathbf G^x$ of the target map which consists of all arrows ending at $x$. A group $G$ can be considered as the groupoid $\mathbf G$ for which there is only one object $\bullet$, so that all arrows are composable, and the fiber $\mathbf G^\bullet$ is just the whole group $G$.

One can easily see that the groupoid moves around the fibers of the target map, namely, given any arrow $\mathbf g$ with the starting point $x$ and the endpoint $y$, one can compose $\mathbf g$ with any arrow whose endpoint is $x$, and the result will be an arrow whose endpoint is $y$. Formally, $$ \mathbf g \mathbf G^x = \mathbf G^y \qquad \forall\,\mathbf g:x\to y \;. $$ One can now talk about the systems $(m^x)$ of measures on the target fibers of a groupoid invariant in the sense that $\mathbf g m^x=m^y$. They are called Haar systems, being a straighforward generalization of the Haar measures in the group case.

In precisely the same way as for groups, there are two ways of relaxing the condition of the existence of a finite Haar system: either to talk about systems of means instead of measures, or to replace exact invariance with an approximate one and to require the existence of a sequence of probability measures $m_n^x$ on the fibers $\mathbf G^x$ such that $$ \| \mathbf g m_n^x - m_n^y \| \to 0 \qquad\,\forall\mathbf g:x\to y \;. $$ In the same way as for groups, these two definitions happen to be equivalent (Renault 1980), and produce what is called amenable groupoids (I skip some technical details here).

Now back to actions. An action of a group $G$ on a space $X$ gives rise to the action groupoid whose objects are the points from $X$, and the arrows are the triples $(x, g ,y)$ with $y=gx$, so that for any $y\in X$ the corresponding fiber $\mathbf G^y$ can be identified just with the group $G$, and the aforementioned action of the groupoid on the fibers of the target map amounts just to the left action of the group on itself. Thus, the amenability of the action groupoid amounts precisely to the existence of a system of means on the group indexed by the action space and invariant with respect to the group action, or, in the approximate terms, to the condition that there exists a sequence of systems $(m_x^n)$ of probability measures on $G$ indexed by the action space, and such that $$ \| g m_x^n - m_{gx}^n \| \to 0 \;. $$
This is precisely the definition from Brown - Ozawa you are asking about (I skip the details concerning the difference between the definitions in the topological and in the measure "worlds").

One more comment. The way the notion of an amenable action was first introduced by Zimmer in 1977 is actually different. His motivation was the fixed point characterization of amenable groups (the existence of a fixed point for any continuous affine action on a compact) - this is the main application of amenability, but is not so convenient for establishing amenability - and given in pretty convoluted terms of existence of invariant sections for certain Banach bundles over the action space (this is the reason some of the papers on amenable action from that period are so excessively long).

For more historical and motivational details see the book by Anantharaman-Delaroche and Renault or a recent preprint of Bühler and Kaimanovich.