Due to complicated rheological behaviors and elastic effect of viscoelastic fluids, only a handful of literatures reporting the large-eddy simulation (LES) studies on turbulent drag-reduction (DR) mechanism of viscoelastic fluids. In addition, these few studies are limited within the low Reynolds number situations. In this paper, LES approach is applied to further study the flow characteristics and DR mechanism of high Reynolds viscoelastic turbulent drag-reducing flow. To improve the accuracy of LES, an N-parallel FENE-P constitutive model based on multiple relaxation times and an improved mixed subgrid-scale (SGS) model are both utilized. DR rate and velocity fluctuations under different calculation parameters are analyzed. Contributions of different shear stresses on frictional resistance coefficient, and turbulent coherent structures which are closely related to turbulent burst events are investigated in details to further reveal the DR mechanism of high Reynolds viscoelastic turbulent drag-reducing flow. Especially, the different phenomena and results between high Reynolds and low Reynolds turbulent flows are addressed. This study is expected to provide a beneficial guidance to the engineering application of turbulent DR technology.