The electronic structure of NaFeAs is studied with angle-resolved photoemission spectroscopy on high quality single crystals. Large portions of the band structure start to shift around the structural transition temperature and smoothly evolve as the temperature lowers through the spin density wave transition. Moreover, band folding due to magnetic order emerges slightly above the structural transition. Our observation provides direct evidence that the structural and magnetic transitions share the same origin and could both be driven by the electronic structure reconstruction in Fe-based superconductors instead of Fermi surface nesting. We did not observe any sign of a gap in the superconducting state, which is likely related to weakened superconductivity in the presence of the spin density wave.