Let $A$ and $R$ be row equivalent $m \times n$ matrices. Let the row vectors of $A$ be $a_1, a_2, a_3, \ldots, a_m$ and the row vectors of $R$ be $r_1, r_2, r_3,\ldots, r_m$. Matrices $A, R$ are row equivalent, therefore the $r$ row vectors are obtained from the $a$ row vectors by elementary row operations. This means that every $r$ row vector is a linear combination of the $a$ row vectors. Therefore the row space of matrix $A$ lies in the row space of matrix $R$.
Is the bolded part in the quote saying $R$ is a subset of row space of $A$? How does that show that row space of $A$ is in the row space of $R$?