The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling ωmax a l-2/3 between the vorticity maximum and the pancake thickness, as was observed in the recent numerical experiments [D. S. Agafontsev et al., "Development of high vorticity structures in incompressible 3D Euler equations," Phys. Fluids 27, 085102 (2015)]. We study the process of pancakes' development in terms of the vortex line representation (VLR), which represents a partial integration of the Euler equations with the explicit conservation of the Cauchy invariants and describes the compressible dynamics of continuously distributed vortex lines. We present, for the first time, the numerical simulations of the VLR equations with high accuracy, which we perform in adaptive anisotropic grids of up to 15363 nodes. With these simulations, we show that the vorticity growth is connected with the compressibility of the vortex lines and find geometric properties responsible for the observed scaling ωmax l-2/3.