Geometric meaning of Koszul modules

Question

Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\mathcal{O}_D)$ is surjective. Kemeny wrote in his paper THE EXTREMAL SECANT CONJECTURE FOR CURVES OF ARBITRARY GONALITY that it is "rather straightforward" that if $L$ is not $p+1$-very ample then the Koszul cohomology $K_{p,2}(C;L)\not=0$. But I cannot see this, at least from the algebraic definition of the Koszul cohomology. Is there any geometric explanation for Koszul cohomology(in which $p+1$-very ampleness might be involved)?