From the PLONK paper.
We describe an optimization by Mary Maller to reduce the number of $F$-elements in the proof from $M$. We begin with an illustrating example. Suppose $V$ wishes to check the identity $h1(X) \cdot h2(X) − h3(X) \equiv 0$. The compilation described above would have $P$ send the values of $h1$, $h2$, $h3$ at a random $x \in F$; and $V$ would check if $h1(x)h2(x)−h3(x) = 0$. Thus, $P$ sends three field elements. Note however, that we could instead have $P$ send only $c := h1(x)$, and then simply verify in the open protocol whether the polynomial $L(X) := c \cdot h2(X) − h3(X)$ is equal to zero at $x$. (Note that we can compute $com(L) = c \cdot com(h2) − com(h3)$.)
Why not just create a new polynomial $F(X) = h1(X) \cdot h2(X) − h3(X)$ & send a commitment to $F(X)$ & also evaluate $F$ at a random point & check if it evaluates to $0$. This will also check the identity needed to be checked. This also reduces number of field elements to be sent. So why a more complicated trick to achieve this?
Another question is about the first quoted line - "proof from $M$" - what or who is $M$ in this context? It's not clear.