Laser manipulation of quantum particles, such as atoms, ions, and molecules, underpins much of modern physics. Electrons, too, can be coherently controlled by light. In this work, we study electron-laser interaction in free space and find that the conventional description based on the effective (ponderomotive) potential requires significant modification. We demonstrate laser-based phase manipulation of the electron wave function by performing interferometric experiments in a transmission electron microscope (TEM) and capture TEM images of the light wave. We then utilize the laser-induced phase shift to realize a nearly ideal phase plate for Zernike phase contrast TEM, solving a long-standing problem and addressing the challenge of dose-efficient interrogation of radiation-sensitive specimens. The laser phase plate is widely expected to advance the TEM studies of protein structure and cell organization.