Electrons and light have to obey certain laws, like conservation of momentum. It may seem strange that light carries momentum, but it's true. Momentum conservation means that motion in a certain direction has to continue in that direction unless there's an outside force to change it. So imagine if a particle of light (a photon) strikes an electron. The electron could absorb the light, but to conserve momentum, the electron then has to start moving in the direction the light was originally traveling. Or the electron could transfer that momentum to something else, like a nearby atom, and start that atom in motion. Or the electron could simply re-emit the light, and the photon of light would continue to carry the momentum in the same, original direction.

The only sign that there had been any interaction between the light and a photon is that light gets slowed down a little bit each time it gets temporarily absorbed and re-emitted. That's what causes refraction, which is why a pencil looks bent when you stick it into a glass of water. Light travels slower through water because it's constantly being absorbed and then re-emitted.