I find using the speed of light confusing as it has little to do with light other than that light happens to travel at that speed [1], but rather the speed of propagation of cause and effect. If you stay absolutely still, cause and effect still propagate at the same speed and so time ticks forward at this speed. If you physically move (have velocity), cause and effect still ticks forward at the same speed, but some of it appears to tick into moving you physically. You can still only propagate at the speed of propagation, but some of that propagation is happening on the physical position axis, so less happens on the time axis.

[1] because light is massless, so in a vacuum there is nothing slowing it down