Light doesn’t have mass, but it is an electromagnetic wave, which means it does interact with anything that has charge, most relevantly including electrons. Basically, when light hits an electron, it gets absorbed and then shot back out in a random direction. Mirrors just have the unique property that when you work out all the interactions it has with light, the only choice light has is to bounce back relative to the direction it came (the self-interaction or reflected light destructively cancels out all light that’s not traveling at an angle equal to the angle of incident).
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