I recently read Why Does E=mc²? by Brian Cox and Jeff Forshaw and in it they explain special relativity. They make a statement about spacetime that I'd not come across before and I found to be a bit of a mind-bender. Everything moves through spacetime (spacetime being of course the combination of the three dimensions of space and the fourth dimension that is time) at the same speed, and that speed is c, speed of light.
This was a totally different way of looking at time-dilation for me and it kind of makes it clear - everything is moving through spacetime at the same speed, but that speed is 'shared between' the spacial dimensions and the time dimension. You may be sat in front of a computer reading this now, stationary. You are not however stationary in spacetime, as clearly you are moving though time. If you treat time as if it were a fourth spacial dimension, that means you are moving through that dimension at c, the speed of light. But what about light? A photon of light (in a vaccuum) is the exact opposite - it is moving through the three spacial dimensions at c, and therefore doesn't travel through the time dimension at all - photons do not experience time passing.
For everything in between, (i.e. things that are moving but at less than the speed of light), the speed is split between space and time, which causes time-dilation. The effect being that the closer you get to travelling through space at c, the slower you will travel through time. The effects are far to small to be noticeable in everyday life and only really come into play when you get pretty near c, but they are there nonetheless.
Hopefully that made sense. Obviously I'm not able to explain it as well as someone like Brian Cox! But the great thing about explanations like that are that as well as making a difficult concept easier to grasp, they also provoke more questions. Like, why does everything move at the same speed through spacetime? Is it even meaningful to imagine something moving at a different speed? In addition to that, photons are massless particles, which is why they move at c through space. So why does possessing the quality we call mass mean you experience time pass? The current theory of mass revolves around the yet to be confirmed Higgs boson and the related Higgs field. Objects passing through the field get slowed down, as if wading through treacle whereas massless things like photons just zip through it unimpeded. Being slowed down by the field manifests itself as having mass. So is there some link between the Higgs field and the dimension of time?