Why $\phi$ $\cdot$ R = $\phi$, rather than $\phi$ $\cdot$ R = R in Automata?
Let us check the definition.

For two sets of strings $S_1$ and $S_2$, the concatenation $S_1\cdot S_2$ consists of all strings of the form $vw$ where $v$ is a string from $S_1$ and $w$ is a string from $S_2$, or formally $S_1\cdot S_2 = \{ vw : v \in S_1, w \in S_2 \}$.

What about $\emptyset\cdot R$?

 Since there is no string in the empty set, we cannot find any string of that form $vw$ where $v$ is a string from the empty set. For example, you cannot form a mixed double in tennis if there is no male players. So $\emptyset\cdot R=\emptyset$.