A neutrino with energy of $sim$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 $sim 3sigma$ level. We monitored the object in the very-high-energy (VHE) band with the MAGIC telescopes for $sim$41 hours 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 $sim 10^{45} - 4 times 10^{46} {rm erg s^{-1}}$. The steep spectrum observed by MAGIC is concordant with internal $gammagamma$ absorption above a few tens of GeV entailed by photohadronic production of a $sim$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 up-scattering 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 co-moving frame can be in the range $sim 10^{14}$ to $10^{18}$ eV.
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