We study interacting fermions in one dimension subject to random, uncorrelated onsite disorder, a paradigmatic model of many‐body localization (MBL). This model realizes an interaction‐driven quantum phase transition between an ergodic and a many‐body localized phase, with the transition occurring in the many‐body eigenstates. We propose a single‐particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one‐particle density matrix (OPDM) in individual many‐body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single‐particle states is a many‐body effect, since without interactions the single‐particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero‐temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation‐spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally‐relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model.