Moni Noar One Way Function: What does input is partioned mean?
I am having trouble understanding what "$m$ is partioned into a sequence $\langle m_1, ..., m_k\rangle$ of elements in $\textbf{Z}_R$ means.


This is essentially what you said. Consider $n$ such that $\{0,1\}^n$, seen as integers, is a subset of $\textbf{Z}_R$ (e.g. $n = \lfloor\log_2R\rfloor$). Then regard $m$ as a long string $m = m_1\|\cdots\|m_N$ where each $m_i$ is in $\{0,1\}^n$ (if $n$ does not divide the bit-length of $m$ then apply a "good" padding to $m$, i.e. one that is prefix-free).

Just take into account that, as noticed by @poncho in the comments, this is not a PRF. This is a 2-universal hash function, and as shown in the post you cite, it is an ingredient for the construction of an actual PRF.