On the hypothesis of the Additive Cousin Problem The Additive Cousin Problem is the following:

Assume that $D$ is a region (open, connected) of $\mathbb{C}^n$. Assume that the Dolbeault Cohomology Group of $D$, $H^1_{\bar{\partial}}(D)$ is equal to zero, and that we have an open covering of $D$ given by $\{ U_j:j\in J\}$. Suppose that we have  meromorphic functions $m_j\in\mathcal{M}(U_j)$ for every $j$ such that, whenever $U_i\cap U_j\neq \emptyset$, $$m_j-m_i\in\mathcal{O}(U_i\cap U_j).$$ Given this, we ask if there is $m\in\mathcal{M}(D)$ such that, in $U_j$, we have $$ m-m_j\in\mathcal{O}(U_j).$$
  Finding such $m$ is called the Additive Cousin Problem.

Can the existence of the $m_j$ meromorphic functions be a consequence of the hypothesis $H^1_{\bar{\partial}}(D)=0$ ? Range's book seems to be implying this, but I do not know how this can be possible.

No, the problem really assumes you have meromorphic functions satisfying certain conditions (perhaps with certain poles). You could always take all the $m_j$ to be constant and then there's no hypotheses at all needed for them to exist. 
Perhaps you already understand how this goes, but if $H^{0,1}_{\bar\partial}(D)=0$, then by the Dolbeault isomorphism we have $H^1(D,\mathcal O) = 0$. Thus, the $\check{\rm C}$ech $1$-cocycle $f_{ij} = m_i-m_j\in Z^1(\scr U,\mathcal O)$ is a coboundary, so we have holomorphic functions $f_i\in \mathcal O(U_i)$ with $f_i-f_j=m_i-m_j$ on $U_i\cap U_j$. Therefore, the functions $f_i-m_i$ glue to give a global meromorphic function $m$.
