...time would stand still for someone traveling at the speed of light.
Considering the speed of light relative to what frame of reference (an appropriate pun in this context)? As Intrepid Spaceman Spiff accelerates to light speed (relative to us) and red-shifts into invisibility, he would seem to stop aging as observed here on Earth. But our inter-galactic hero experiences ordinary aging in his own frame of reference.
It all follows from there being no special reference point in space and time from which all measurements are made, and the total energy of the universe being independent of where you are and how fast you're moving when you measure it.
The models of Special and General Relativity seem strange and counter-intuitive, but they work, which is why they are accepted.
Project Gutenberg has an online copy of Einstein's own explanation, which is comprehensible to non-experts.