A neutrino with energy ˜290 TeV, IceCube-170922A, was detected in coincidence with the BL Lac object TXS 0506+056 during enhanced gamma-ray activity, with chance coincidence being rejected at ˜3σ level. We monitored the object in the very-high-energy (VHE) band with the Major Atmospheric Gamma-ray Imaging Cherenkov (MAGIC) telescopes for ˜41 hr from 1.3 to 40.4 days after the neutrino detection. Day-timescale variability is clearly resolved. We interpret the quasi-simultaneous neutrino and broadband electromagnetic observations with a novel one-zone lepto-hadronic model, based on interactions of electrons and protons co-accelerated in the jet with external photons originating from a slow-moving plasma sheath surrounding the faster jet spine. We can reproduce the multiwavelength spectra of TXS 0506+056 with neutrino rate and energy compatible with IceCube-170922A, and with plausible values for the jet power of ˜ {10}45-4× {10}46 {erg} {{{s}}}-1. The steep spectrum observed by MAGIC is concordant with internal γγ absorption above ˜100 GeV entailed by photohadronic production of a ˜290 TeV neutrino, corroborating a genuine connection between the multi-messenger signals. In contrast to previous predictions of predominantly hadronic emission from neutrino sources, the gamma-rays can be mostly ascribed to inverse Compton upscattering of external photons by accelerated electrons. The X-ray and VHE bands provide crucial constraints on the emission from both accelerated electrons and protons. We infer that the maximum energy of protons in the jet comoving frame can be in the range ˜1014 - 1018 eV.