We study the valence band structure of ReSe$_{2}$ crystals with varying thickness down to a single layer using nanoscale angle-resolved photoemission spectroscopy and density functional theory. The width of the top valence band in the direction perpendicular to the rhenium chains decreases with decreasing number of layers, from 280 meV for the bulk to 61 meV for monolayer. This demonstrates increase of in-plane anisotropy induced by changes in the interlayer coupling and suggests progressively more one-dimensional character of electronic states in few-layer rhenium dichalcogenides.