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$\ce{H2O}$ molecule is dissociated into $\ce{OH-}$ and $\ce{H+}$ ions on reduction at the cathode in electrolysis. But what happens at the atomic level to $\ce{H2O}$ that proton and hydroxide ions get separated after gaining electrons from cathode? What really happens when water molecule gains electrons from cathode?
$$\ce{2H2O(l) + 2 e- -> H2(g) + 2 OH-(aq)}$$
The dissociation of water to $\ce{OH-}$ and $\ce{H+}$ happens first before any interaction with the cathode. $\ce{H+}$ ions then gain an electron at the cathode to produce hydrogen. At the anode, water is oxidized (loses electrons) to produce oxygen and more $\ce{H+}$ ions.
In pure water there is a very small concentration of $\ce{H+}$ and $\ce{OH-}$ which is why the reaction is very slow and not thermodynamically favorable. It is also the reason why pure water is such a good insulator. But when you add an electrolyte the process becomes much more favourable kinetically and thermodynamically. Further proof that $\ce{H+}$ and $\ce{OH-}$ are generated before any interaction with the cathode.
