Why does light move?
Now, light travels at the speed of light, so if we consider a photon as the reference point, isn't time still for it? And if time is stopped, how does the photon ever get anywhere?

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Of course, this whole argument is just juvenile sophistry.
Light moves because a time varying electric field induces a time varying magnetic field, which induces an electric field, and so on. The wave equation you can derive from Maxwell's equations includes a numerical constant which specifics the speed of the wave. Having such a numerical constant is inconsistent with Newtonian mechanics (technically, it is inconsistent with the Galilean rules for velocity transformation).
Einstein's insight was that Maxwell's equations are right, and the Galilean transforms are wrong. He showed that using the Lorentz transformations for velocity yields Maxwell's equations for light, and in the limit of low velocities, reduces to the familiar Galilean result. It is only in the intermediate regime (velocities high compared with normal experience, fractions of c) that the Lorentz transformations give unexpected results.
Remember that a photon is an abstract-concept that is useful for some situations (when you're digesting Kelsey's excellent answer)
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That is also why it moves so fast.