We report dissipative magnon-photon coupling caused by the cavity Lenz effect, where the magnons in a magnet induce a rf current in the cavity, leading to a cavity backaction that impedes the magnetization dynamics. This effect is revealed in our experiment as level attraction with a coalescence of hybridized magnon-photon modes, which is distinctly different from level repulsion with mode anticrossing caused by coherent magnon-photon coupling. We develop a method to control the interpolation of coherent and dissipative magnon-photon coupling, and observe a matching condition where the two effects cancel. Our work sheds light on the so-far hidden side of magnon-photon coupling, opening a new avenue for controlling and utilizing light-matter interactions.