$k_i$ is the number of ways to make the $i^{th}$ choice. So for example, if there are 6 ways to make the second choice, then we have $k_2=6$.
The question is asking how many ways we can make all of our choices if we can make some number of choices, $k_i$ in this case, the $i^{th}$ time we need to make a choice. Assuming none of these choices influence the others, then we have
$k_1k_2\cdots k_n=\prod_{i=1}^{n} k_i$ total ways to make all of our choices.
An example: if we need to make a sandwich, picking a sandwich bread from either rye, white, or wheat, picking a meat from beef, turkey, ham, or chicken, and picking a cheese from cheddar or pepperjack, we would have $k_1=3$, $k_2=4$, and $k_3=2$. This gives us a total of $k_1k_2k_3=3\cdot 4\cdot2=24$ ways to make a sandwich.