Let $M(2,d)$ be the set of all vector bundles of rank $2$ and degree $d$ over a smooth projecitve curve of genus $g\geq 3$. Let $M(2,0)^s$ and $M(2,0)^{ss}$ be the stable and semistable vector bundles of $M(2,0)$ respectively.
Let $E \in M(2,0)^{ss}- M(2,0)^s$, then E can be written as $L_1\oplus L_2$ where $L_1$ and $L_2$ are line bundles of degree $\frac{d}{2}$.
Question: Why is this is true?
My attempt- We can find a line sub-bundle of L of E. $L/E$ also will be a line bundle.
So we have an exact sequence $0\rightarrow L\rightarrow E \rightarrow L/E \rightarrow 0$.
Is this exact sequence splits, so that we can write $E =L \oplus L/E$.
If I assume that it splits, how did we get the degree $\frac{d}{2}$ of the line bundles $L$ and $L/E$
I think you mean $M(2, d)$ is the set of all semi-stable rank 2 bundle of degree $d$ (not just ${\bf all}$ rank 2 vector bundle of degree $d$ because in this case it is trivially not true, for example, take $E = \mathcal{O}_{\mathbb{P}^1}(d - 1) \oplus \mathcal{O}_{\mathbb{P}^1}(d + 1)$ over $\mathbb{P}^1$) and everything is on a smooth projective curve $C$.
EDIT: (using your new notations) $E \in M(2, d)^{ss} - M(2, d)^s \Rightarrow $ there exists a line sub-bundle $L$ of $E$ whose degree is equal to $\mu(E) = \frac{d}{2}.$ So the quotient $M := E/L$ is also a line bundle with degree $\frac{d}{2}.$ Now we have an exact sequence $0 \rightarrow L \rightarrow E \rightarrow M \rightarrow 0$ of vector bundles. These extensions are classified by $H^1(C, M^{-1} \otimes L) \cong H^0(C, M \otimes L^{-1} \otimes K_C),$ where $K_C$ is the caninocal bundle of $C.$ But I don't see why the above exact sequence will corresponds to the trivial element of $H^0(C, M \otimes L^{-1} \otimes K_C),$ in general.
${\bf EDIT :}$ Your claim is not true in general. Because the strictly semi-stable rank 2 bundles over an elliptic curve $C$ are either of the form $L \oplus M,$ where deg $L =$ deg $M$, or of the form $\mathcal{E} \otimes L,$ where $L$ is a line bundle on $C$ and $\mathcal{E}$ is the unique non-split extension of $\mathcal{O}_C$ by $\mathcal{O}_C$. (using the classification of rank 2 bundles over elliptic curve, due to Atiyah.)
${\bf EDIT:}$ Let $E$ be a rank $2$ strictly semi-stable bundle of degree $d$ over a smooth projective curve $C$. Then $E$ can be wriiten as an extension $$ 0 \rightarrow L \rightarrow E \rightarrow M \rightarrow 0 $$ where $L, M$ are line bundles on $C$ and deg $L = \frac{d}{2} =$ deg $M,$ and $M = L^* \otimes det(E)^{-1}.$ Such extensions are classified by the group $H^1(C, M^{-1} \otimes L).$ If $H^1(C, M^{-1} \otimes L) \neq 0,$ then there is a non-split extension. On the other hand, any such extension is a Jordan-Holder filtration for the bundle $E.$ So up to $S$-equivalence, we can say that $E$ is a direct sum of line bundles $L \oplus M,$ where $L, M$ are line bundles on $C$ and deg $L = \frac{d}{2} =$ deg $M.$