For globally generated line bundles it is quite clear to see that they are nef: Let $L$ be globally generated, then the global sections in $H^0(X,L)$ define a morphism $\Phi \colon X \to \mathbb{P}_N$, such that $L = \Phi^*(\mathcal{O}_{\mathbb{P}_N}(1))$. The line bundle $\mathcal{O}_{\mathbb{P}_N}(1)$ is ample which imples nef, and the pullback is nef as well, hence $L$ is nef.
How do we obtain this for vector bundles $E$ as well, such that if $E$ is globally generated it is nef?
Thanks in advance.
Let me make my comment an answer for completeness.
$O(1)$ is the tautological quotient of $\pi^* E$. If $E$ is globally generated, then so is $\pi^* E$, hence so too is $O(1)$. Since it's a line bundle, that implies it's nef.