Given a complex-analytic manifold of dimension $d$, why does the cohomology of coherent sheaves vanish in dimension $> 2d$ (without using GAGA)?
Note that it is not necessary to say to avoid GAGA, as GAGA has no relevance in the absence of compactness assumptions.
Anyway, something much more general (and satisfying) is true: all topological sheaf cohomology on a (paracompact Hausdorff) analytic space vanishes beyond twice the analytic dimension. Here is a sketch of a proof. By metrization theorems, such spaces are metrizable. Also, connected components of analytic spaces are open, so cohomology is direct product of cohomologies on the connected components. We can therefore restrict attention to the connected case, so the underlying topological space is separable (i.e., countable base of opens); I think this latter fact is stated with reference in the Introduction of the book Theory of Stein Spaces. (If not, assume the given analytic space is separable, a very "practical" assumption!)
By the local analytic "Noether normalization" (really Weierstrass Preparation), twice the analytic dimension equals the "topological dimension" in the sense of dimension theory as in Engelking's marvelous book "General topology" for separable metric spaces. (For separable metric spaces, various notions of topological dimension are proved to agree; all done in that book. For opens in a real Euclidean space it recovers the expected "dimension"!) That book shows open covers of separable metric spaces have refinements whose $(n+1)$-fold overlaps are empty for $n$ beyond the topological dimension (in one of the various equivalent senses of dimension: the "covering" dimension!). Now use equality of Cech and derived functor cohomology for paracompact Hausdorff spaces to conclude.