Haven't had one of these for awhile. This thread is for questions or comments that you've felt silly about not knowing/understanding. Let's try to exchange info that seems obvious, knowing that due to the illusion of transparency it really isn't so obvious!
Other people have explained this pretty well already, but here's a non-rigorous heuristic that might help. What follows is not technically precise, but I think it captures an important and helpful intuition.
In relativity, space and time are replaced by a single four-dimensional space-time. Instead of thinking of things moving through space and moving through time separately, think of them as moving through space-time. And it turns out that every single (non-accelerated) object travels through space-time at the exact same rate, call it c.
Now, when you construct a frame of reference, you're essentially separating out space and time artificially. Consequently, you're also separating an object's motion through space-time into motion through space and motion through time. Since every object moves through space-time at the same rate, when we separate out spatial and temporal motion, the faster the object travels through space the slower it will be traveling through time. The total speed, adding up speed through space and speed through time, has to equal the constant c.
So an object at rest in a particular frame of reference has all its motion along the temporal axis, and no motion at all along the spatial axes. It's traveling through time at speed c and it isn't traveling through space at all. If this object starts moving, then some of the temporal motion is converted to spatial motion. It's speed through space increases, and its speed through time decreases correspondingly, so that the motion through space-time as a whole remains constant at c. This is the source of time dilation in relativity (as seen in the twin paradox) - moving objects move through time more slowly than stationary objects, or to put it another way, time flows slower for moving objects.
Of course, the limit of this is when the object's entire motion through space-time is directed along the spatial axes, and none of it is directed along the temporal axes. In this case, the object will move through space at c, which turns out to be the speed of light, and it won't move through time at all. Time will stand still for the object. This is what's going on with photons.
From this point of view, there's nothing all that weird about a photon's motion. From the space-time perspective, which after all is the fundamental perspective in relativity, it is moving pretty much exactly like any other object. It's only our weird habit of treating space and time as extremely different that makes the entirely spatial motion of a photon seem so bizarre.
That is helpful, and interesting, though I think I remain a bit confused about the idea of 'moving through time' and especially 'moving through time quickly/slowly'. Does this imply some sort of meta-time, in which we can measure the speed at which one travels through time?
And I think I still have my original question: if a photon travels through space at c, and therefore doesn't travel through time at all, is the photon at its starting and its final position at the same moment? If so, in what sense did it travel through space at all?