Seemingly unrelated empirical hydrologic laws and several experimental facts related to the fractal geometry of the river basin are shown to find a natural explanation into a simple finite-size scaling ansatz for the power laws exhibited by cumulative distributions of river basin areas. Our theoretical predictions suggest that the exponent of the power law is directly related to a suitable fractal dimension of the boundaries, to the elongation of the basin, and to the scaling exponent of mainstream lengths. Observational evidence from digital elevation maps of natural basins and numerical simulations for optimal channel networks are found to be in good agreement with the theoretical predictions. Analytical results for Scheidegger's trees are exactly reproduced.