Why when I travel at near light speed and return to Earth has more time passed there than for me?

I have heard many times if I leave Earth and travel at near light speed, time will slow down so that the speed of light will continue to be perceived as constant to me. I understand that concept.

I have heard many times that if I return to Earth, they will have experienced the ‘real’ passage of time, and they will be much older. That I do not get and I would ask someone to guide me to understanding.

The sticking point is as follows: Hasn’t the Earth, at least while I was heading away from that planet, been travelling at near light speed from my perspective? Why wouldn’t our clocks synchronize when I landed back on Earth.

Thank you for your patience; my failure to grasp this has been driving me nuts for a long time.