We study spectral and wave-function statistics for many-body localization transition in systems with long-range interactions decaying as 1/rα with an exponent α satisfying d≤α≤2d, where d is the spatial dimensionality. We refine earlier arguments and show that the system undergoes a localization transition as a function of the rescaled disorder W∗=W/L2d-αlnL, where W is the disorder strength and L the system size. This transition has much in common with that on random regular graphs. We further perform a detailed analysis of the inverse participation ratio (IPR) of many-body wave functions, exploring how ergodic behavior in the delocalized phase switches to fractal one at the critical point and on the localized side of the transition. Our analytical results for the scaling of the critical disorder W with the system size L and for the scaling of IPR in the delocalized and localized phases are supported and corroborated by exact diagonalization of spin chains.