If $M$ is an $R$-module and $I$ is an ideal of $R$, then $IM$ is a submodule of $M$, and there is a natural $R/I$ module structure on $M/IM$: namely, given $r+I\in R/I$ and $m+IM$ in $M/IM$, define $(r+I)(m+IM) = rm+IM$.
This is well-defined: if $r-s\in I$ then $(r-s)m\in IM$, so $rm+IM = sm+IM$. And if $n-m\in IM$, then we can express $n-m$ as a sum $n-m = a_1m_1+\cdots +a_km_k,\qquad a_i\in I, m_j\in M$ so $r(n-m) = ra_1m_1+\cdots + ra_km_k\in IM$ since $ra_i\in I$ for all $i$.
Here, you have $M=m_1\cdots m_{k-1}$, $I=m_k$, so there is a natural $R/m_k$ module structure on $M/IM = m_1\cdots m_{k-1}/m_1\cdots m_k$. Since $m_k$ is maximal, $R/m_k$ is a field, so this is actually a vector space structure. So this is a natural field over which to give $M/IM$ a vector space structure, but by no means the only one.