I am not a Physicist, so please don't rely on this, but my understanding is that this comes down to the frames of reference in special relativity.

If you're in a car on the highway, an observer standing still outside the car sees you moving at 50mph, but an observer inside the car with you, sees you not moving at all, but the world rushing past at 50mph. These are two different reference frames, and the second one is called a rest frame - any object can be considered to have a rest frame of reference where it isn't moving at all. Any object that is, which has mass.

Massless objects are always observed to be moving at the speed of light by any observer in any frame of reference. This is one of the most mind-bending parts of relativity, but it ultimately connects to the part of special relativity where speed affects time - the faster you go, the slower your clock runs compared to things moving slower than you, and the closer you get to the speed of light, the closer your clock gets to zero. Massless particles only travel at the speed of light, and thus have no clock, therefore they experience no time.

I hope I got that right, but even if I did, I still don't feel like I actually understand it at all!