Just to make it clear: in this context "smash product" of $i$ and $j$ means the map $$ i\square j \colon (A\times B')\amalg_{A\times B} (A'\times B)\to A'\times B' $$ constructed from $i\colon A\to A'$ and $j\colon B\to B'$. (I can't find a place in the book where Jacob defines "smash product" properly, though this is how he uses it.)
The first observation is that "smash product" is associative: $$ (i\square j)\square k \approx i\square (j\square k). $$ The second observation is that it is commutative: $i\square j\approx j\square i$.
The third observation is that the smash product of two monomorphisms is a monomorphism (=cofibration).
Fourth observation: if a "weakly saturated" class $C$ of maps (such as inner anodynes) is generated by a class $G$, then to show $C\square j\subseteq C$ it suffices to show $G\square j\subseteq C$.
To show that if $i$ is inner anodyne and $j$ a monomorphism, then $i\square j$ is inner anodyne, it is enough to show it for $i$ in a generating class for inner anodynes, e.g., $i$ in $A_3$. So if $i=i'\square u$ with $i'$ a monomorphism and $u\colon \Lambda^1_1\to \Delta^2$, we get $$ i\square j = (i'\square u)\square j \approx (i'\square j)\square u, $$ which is in $A_3$.
Added. About the fourth observation: given a class of maps $S$, write $\overline{S}$ for the weak saturation of $S$, i.e., the class containing $S$ and closed under the various constructions that weak saturated classes are supposed to be closed under. Then in general, we have that $$ \overline{S}\square j \subseteq \overline{S\square j}.$$
You can prove this directly from the definition of weak saturated class. A different argument uses the idea that $\overline{S\square j}$ is exactly the class of maps which have the left-lifting-property with respect to the class of maps with the right-lifting-property with respect to $S\square j$ (this uses the small-object argument). I don't feel like writing it up here: see for instance section 15 in my notes on quasicategories: http://www.math.uiuc.edu/~rezk/quasicats.pdf