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 $L_1=1$, then any section $\phi\in\Gamma(E)$ has $div(\phi)$, 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, which maybe you should have explained in the question 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.