Context. While classical models successfully reproduce intensities of many transition region lines, they predict helium extreme-UV (EUV) line intensities roughly an order of magnitude lower than the observed value. Aims. Our aim is to determine the relevant formation mechanism(s) of the helium EUV resonance lines capable of explaining the high intensities under quiet Sun conditions. Methods. We synthesised and studied the emergent spectra from a 3D radiation-magnetohydrodynamics simulation model. The effects of coronal illumination and non-equilibrium ionisation of hydrogen and helium are included self-consistently in the numerical simulation. Results. Radiative transfer calculations result in helium EUV line intensities that are an order of magnitude larger than the intensities calculated under the classical assumptions. The enhanced intensity of HeI lambda 584 is primarily caused by He II recombination cascades. The enhanced intensity of He II lambda 304 and He II lambda 256 is caused primarily by non-equilibrium helium ionisation. Conclusions. The analysis shows that the long standing problem of the high helium EUV line intensities disappears when taking into account optically thick radiative transfer and non-equilibrium ionisation effects.