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We report resolved near-infrared spectroscopy and photometry of the recently identified brown dwarf binary WISE J104915.57-531906.1AB, located 2.02±0.15 pc from the Sun. Low-resolution spectral data from Magellan/FIRE and IRTF/SpeX reveal strong H₂ O and CO absorption features in the spectra of both components, while the secondary also exhibits weak CH₄absorption at 1.6 µm and 2.2 µm. Spectral indices and comparison to low-resolution spectral standards indicate component types of L7.5 and T0.5±1, the former consistent with the optical classification of the primary. Both sources also have unusually red spectral energy distributions for their spectral types, which we attribute to enhanced condensate opacity (thick clouds). Relative photometry reveals a flux reversal between the J and K bands, with the T dwarf component being brighter in the 0.95-1.3 µm region (ΔJ = -0.31 ± 0.05). As with other L/T transition binaries, this reversal likely reflects the depletion of condensate opacity in the T dwarf, with the contrast enhanced by the thick clouds present in the photosphere of the L dwarf primary. The 1 µm flux from the T dwarf most likely emerges from gaps in its cloud layer, as suggested by the significant optical variability detected from this source by Gillon et al. Component mass measurements of the WISE J1049-5319AB system through astrometric and component radial velocity monitoring may resolve the current debate as to whether the loss of photospheric condensate clouds at the L dwarf/T dwarf boundary is a slow or rapid process, a conceivable endeavor given its proximity, brightness, small separation (3.1±0.3 AU), and reasonable orbital period (20-30 yr).