The extragalactic background light (EBL) is the radiation accumulated through the history of the Universe in the wavelength range from the ultraviolet to the far infrared. Local foregrounds make the direct measurement of the diffuse EBL notoriously difficult, while robust lower limits have been obtained by adding up the contributions of all the discrete sources resolved in deep infrared and optical galaxy observations. Gamma-ray astronomy has emerged in the past few years as a powerful tool for the study of the EBL: very-high-energy (VHE) photons traversing cosmological distances can interact with EBL photons to produce e+e- pairs, resulting in an energy-dependent depletion of the gamma-ray flux of distant sources that can be used to set constraints on the EBL density. The study of the EBL is one of the key scientific programs currently carried out by the MAGIC collaboration. We present here the results of the analysis of 32 VHE spectra of 12 blazars in the redshift range 0.03-0.94, obtained with over 300 hours of observations with the MAGIC telescopes between 2010 and 2016. A combined likelihood maximization approach is used to evaluate the density and spectrum of the EBL most consistent with the MAGIC observations. The results are compatible with state-of-the-art EBL models, and constrain the EBL density to be within ≈ 20% the nominal value in such models. The study reveals no anomalies in gamma-ray propagation in the large optical depth regime - contrary to some claims based on meta-analyses of published VHE spectra.