Recent high-resolution observations with the 1 m Swedish Solar Telescope (SST) on La Palma reveal the fine substructure of penumbral filaments and new properties of their dynamics. These findings provide both the basis and constraints for novel models of the penumbra. We present new observations of a large isolated sunspot near Sun center obtained with the SST in 2006. Our data, taken simultaneously in the 4305 A G-band and 4364 A continuum bandpasses and compiled in high-cadence movies, confirm the previous results and reveal new features of penumbral filament dynamics. We find that individual filaments are cylindrical helices with the apparent properties of vortex tubes exhibiting flow patterns similar to kinked flux tubes. Measured pitch/radius ratios of helical filaments indicate their dynamic stability. We propose a mechanism that explains the fine structure of penumbral filaments, their observed dynamics, and their formation process in association with sunspot properties. The mechanism assumes that the umbra itself is a dense conglomerate of twisted, interlaced flux tubes with peripheral filaments branching out from the trunk at different heights due to ongoing reconnection processes and arcing downward to the photosphere. The twist of individual filaments, and the resulting distribution of magnetic fields and temperature, is due to the onset of the well-known screw pinch instability, the parameters of which can be measured from our data.