We present spectrophotometry spanning 1–5 μm of 51 Eridani b, a 2–10 M_(Jup) planet discovered by the Gemini Planet Imager Exoplanet Survey. In this study, we present new K1 (1.90–2.19 μm) and K2 (2.10–2.40 μm) spectra taken with the Gemini Planet Imager as well as an updated L_P (3.76 μm) and new M_S (4.67 μm) photometry from the NIRC2 Narrow camera. The new data were combined with J (1.13–1.35 μm) and H (1.50–1.80 μm) spectra from the discovery epoch with the goal of better characterizing the planet properties. The 51 Eri b photometry is redder than field brown dwarfs as well as known young T-dwarfs with similar spectral type (between T4 and T8), and we propose that 51 Eri b might be in the process of undergoing the transition from L-type to T-type. We used two complementary atmosphere model grids including either deep iron/silicate clouds or sulfide/salt clouds in the photosphere, spanning a range of cloud properties, including fully cloudy, cloud-free, and patchy/intermediate-opacity clouds. The model fits suggest that 51 Eri b has an effective temperature ranging between 605 and 737 K, a solar metallicity, and a surface gravity of log(g) = 3.5–4.0 dex, and the atmosphere requires a patchy cloud atmosphere to model the spectral energy distribution (SED). From the model atmospheres, we infer a luminosity for the planet of −5.83 to −5.93 (log L/L_⊙), leaving 51 Eri b in the unique position of being one of the only directly imaged planets consistent with having formed via a cold-start scenario. Comparisons of the planet SED against warm-start models indicate that the planet luminosity is best reproduced by a planet formed via core accretion with a core mass between 15 and 127 M_⊕