Particle that only has mass when moving in one direction observed for first time

“A particle can have no mass when its energy is entirely derived from its motion, meaning it is essentially pure energy traveling at the speed of light.”

I never understood this. In this case, initially, it is assumed that mass is a property of matter of this moving body. Then we assume that mass disappears but matter or body keeps moving without mass. It looks like there is equivocation or some other kind of play on words here. Can anyone clarify without using the phrase “in Einstein’s special relativity. . .”?