We report the first-principles study of the correlated behavior of the magnetic anisotropy energy (MAE) and orbital moment anisotropy (OMA) as the functions of the thickness N of the Fe film. The work is motivated by recent experimental studies combining photoemission, x-ray magnetic circular dichroism, and magnetic anisotropy measurements. In agreement with experiment, the correlated oscillations of MAE(N) and OMA(N) are obtained that have their origin in the formation of the 3d quantum well states (QWS) confined in the films. The main contribution to the oscillation amplitude comes from the surface layer. This is an interesting feature of the phenomenon consisting in the peculiar dependence of the physical quantities on the thickness of the film. We demonstrate that the band structure of the bulk Fe does not reflect adequately the properties of the 3d QWS in thin films and, therefore, does not provide the basis for understanding the oscillations of MAE(N) and OMA(N). A detailed point-by-point analysis in the two-dimensional (2D) Brillouin zone (BZ) of the film shows that the contribution of the Γ point, contrary to a rather common expectation, does not play an important role in the formation of the oscillations. Instead, the most important contributions come from a broad region of the 2D BZ distant from the center of the BZ. Combining symmetry arguments and direct calculations we show that orbital moments of the electronic states possess nonzero transverse components orthogonal to the direction of the spin magnetization. The account for this feature is crucial in the point-by-point analysis of the OMA. On the basis of the calculations for noncollinear spin configurations we suggest interpretations of two interesting experimental findings: fast temperature decay of the oscillation amplitude in MAE(N) and unexpectedly strong spin mixing of the initial states of the photoemission process.