Taken this excerpt directly from Zorich, Mathematical Analysis II:
My question is simply this: I'd like to understand what the author means by the last sentence highlighted in red, which seems to me to seem to be in contrast with the first (always highlighted in red).
It's just the notation that's unfortunate here. I haven't seen this done with Riemann integrals, but with Lebesgue integrals, the vector space of Lebesgue integrable functions on $E$ is denoted $\mathcal L(E)$, while the quotient space in which almost everywhere equal functions are identified is denoted $L(E)$. I'd suggest doing the same here: Denote the space of Riemann integrable functions as $\mathcal R(E)$, and the quotient space in question as $R(E)$, so they are easily distinguished.
Now what Zorich says is essentially just that while the integral is linear as a function $\mathcal R(E)\longrightarrow\mathbb R$, it is also linear as a function $R(E)\longrightarrow\mathbb R$, where $[f]\mapsto\int_E[f]\mathrm dV:=\int_E f\mathrm dV$. Here, $[f]$ is the equivalence class of $f$ in the quotient space.