The low-temperature properties of amorphous solids are widely believed to be controlled by the low-frequency quasilocalized modes. However, what governs their spatial structure and density is unclear. We study these questions numerically in very large systems as the jamming transition is approached and the pressure p vanishes. We find that these modes consist of an unstable core and a stable far-field component. The length scale of the core diverges as p(-1/4) and its characteristic volume diverges as p(-1/2). These spatial features are precisely those of the anomalous modes that are known to cause the boson peak in the vibrational spectra of weakly coordinated materials. From this correspondence, we deduce that the density of quasilocalized modes must follow g(loc) (omega) similar to omega(4)/p(2), which is in agreement with previous observations. Thus, our analysis demonstrates the nature of quasilocalized modes in a class of amorphous materials.