The relation between the two is that they agree, but actually more can be said. The variety $Z$ in your notation is naturally a $\mathbb{G}_m$ torsor over $X$, so that there is a free action of the multiplicative group on the total space $Z$ of this torsor. The equation $\Gamma(Z, \mathcal{O}_Z) \cong \oplus_k H^0(X,L^k)$ is then an isomorphism of graded rings ,where the right hand side is graded by $-k$ and the left hand side by the grading induced from the $\mathbb{G}_m$-action. Explicitly, the $k$-th piece on the left hand side consist of functions $f$ on $Z$ such that $f(tx) = t^k f(x)$ for $t \in \mathbb{G}_m$.
To see why this isomorphism holds, one only has to identify fucntions on $Z$ which are homogeneous of degree $k$ with $L^{-k}$. For $k = 1$ this follows from the more or less tautological fact that linear functionals (i.e. homogeneous functions of degree 1) along the fibers of $L$ agree with sections of $L^* = L^{-1}$. The rest of the $k$-s are similar, just note that unctions which are homogeneous of degree $k$ along the fibers form the sections of a line bundle, and that locally each such function is a product of $k$ functions of degree 1.
