I'm pretty sure that by $L_i>1$ he means that the line bundle has sections. Now this either means that $L_i=1$ or $\deg L_i\geq 1$. On an elliptic curve, any line bundle (divisor) of positive degree is non-special so for any of these $L_i$ with $\deg L_i\geq 1$, we must have $H^1(L_i)=0$. Also in general on a curve, any line bundle $L$ of degree 0 with a section is precisely 1. So this answers your other questions. About why $L_i>1$, if you look above in the paper at Lemma 10, then you see that if $L_1=1$, where $L_1$ is by definition the sub line bundle of maximal degree, then any section $\phi\in\Gamma(E)$ has $div(\phi)$$div(\phi)=0$, and therefore $\Gamma(E)$ generates a trivial sub bundle $\mathcal O^s$, where $s=\dim\Gamma(E)$. But as $s\geq d$ by Riemann-Roch, this is a contradiction to $d\geq r$. From the definition of maximal splitting, whichthe line bundles $L_i$ are constructed inductively, so $L_i\geq L_{i-1}$, and thus we see that all of the $L_i$ have $h^1=0$. Hope this helps. Also, maybe you should have explained in the question what the maximal splitting is since the vector bundles you're talking about are indecomposable and thus it's not a real splitting. Since $L_i\geq L_{i-1}$, we see that all of the $L_i$ have $h^1=0$. Hope this helps.
