We report on the discovery of three transiting super-Earths around K2-155 (EPIC 210897587), a relatively bright early M dwarf (V = 12.81 mag) observed during Campaign 13 of the NASA K2 mission. To characterize the system and validate the planet candidates, we conducted speckle imaging and high-dispersion optical spectroscopy, including radial velocity measurements. Based on the K2 light curve and the spectroscopic characterization of the host star, the planet sizes and orbital periods are 1.55(-0.17)(+0.20) R-circle plus and 6.34365 +/- 0.00028 days for the inner planet; 1.95(-0.22)(+0.27) R-circle plus and 13.85402 +/- 0.00088 days for the middle planet; and 1.64(-0.17)(+0.18) R-circle plus and 40.6835 +/- 0.0031 days for the outer planet. The outer planet (K2-155d) is near the habitable zone, with an insolation 1.67 +/- 0.38 times that of the Earth. The planet's radius falls within the range between that of smaller rocky planets and larger gas-rich planets. To assess the habitability of this planet, we present a series of three-dimensional global climate simulations, assuming that K2-155d is tidally locked and has an Earth-like composition and atmosphere. We find that the planet can maintain a moderate surface temperature if the insolation proves to be smaller than similar to 1.5 times that of the Earth. Doppler mass measurements, transit spectroscopy, and other follow-up observations should be rewarding, as K2-155 is one of the optically brightest M dwarfs known to harbor transiting planets.