Context. The BL Lac object S5 0716+714, a highly variable blazar, underwent an impressive outburst in January 2015 (Phase A), followed by minor activity in February (Phase B). The MAGIC observations were triggered by the optical flux observed in Phase A, corresponding to the brightest ever reported state of the source in the R-band. Aims.The comprehensive dataset collected is investigated in order to shed light on the mechanism of the broadband emission. Methods. Multi-wavelength light curves have been studied together with the broadband spectral energy distributions (SEDs). The sample includes data from Effelsberg, OVRO, Metsähovi, VLBI, CARMA, IRAM, SMA, Swift-UVOT, KVA, Tuorla, Steward, RINGO3, KANATA, AZT-8+ST7, Perkins, LX-200, Swift-X...