It's very weird to someone who has never been exposed to this before, who has only ever worked with vectors in a context where they certainly can be of varying magnitudes, like velocity normally is in Newtonian mechanics.
That's because the four velocity is not really a velocity. The path of a particle through spacetime can be described by a function p(r) = (x(r),y(r),z(r),t(r)) that gives the position (x,y,z,t) as a function of a parameter r. This way the point p(r) traces out some path in spacetime.
Now, the same path is described by many different parameterizations. For example, if we pick q(r) = p(2r) that's the same path. The point varies twice as fast with r, but that has no physical meaning. The total path that is traced out in spacetime is still the same path. That is, the set {p(r) | r in [-∞,∞]} is the same as the set {q(r) | r in [-∞,∞]}, and it is only this set that has physical meaning.
In fact, we can also pick q(r) = p(r^3 + r) or some other parameter transformation.
We might look at the vector p'(r) tangent to the path. The length of this vector depends on the parameterization, but its direction doesn't. So p'(r)/|p'(r)| is a true physical property of the path, but the length |p'(r)| is just an artefact of the parameterization that we chose to describe the path.
In order to make the parameterization somewhat more canonical, we usually pick a parameterization where |p'(r)| = c. As it happens, the four velocity is defined to be p'(r) given this choice of parameterization.
So the length of the four velocity is c simply by definition. This statement has no physical meaning. The statement "we travel through time with the speed of light" is uninteresting. It tells you nothing about physics. It only tells you something about the conventions we use when we describe paths through spacetime. We could very well have picked the convention |p'(r)| = c/2, and then we'd have the statement "we travel through time with half the speed of light", or even |p'(r)| = 2c, and then we'd "travel through time with double the speed of light".
Given any path through spacetime whatsoever, we can pick a parameterization such that it travels through spacetime at the speed of light. So the condition that something travels through spacetime with the speed of light places no constraints on the path that it takes. The statement "we travel through spacetime with the speed of light" makes it sound like there are paths that do travel through spacetime with the speed of light, and there are paths that do not travel through spacetime with the speed of light, and physical particles go along paths of the first kind. This is wrong. There are no paths that do not travel through spacetime at the speed of light, because whether or not it does is a property of the parameterization that we choose to describe the path, not a property of the path itself.
This isn't wrong, but also somewhat misses the point.
Four-velocity is change of spatio-temporal position (Δx,Δt) in an inertial frame per unit of time Δτ measured by the object in motion. For a concrete example, think a race track with distance markers accompanied by (synchronized) stationary clocks, and a vehicle carrying its own clock. Three-velocity will be given by Δx/Δt (the distance along the track divided by the time on the stationary clock), whereas the spatial component of four-velocity will be given by Δx/Δτ (the distance along the track divided by the time on the vehicle's clock). The temporal component of four-velocity will be c·Δt/Δτ, ie proportional to the clock ratio.
Turns out no matter the speed of the vehicle, it will always hold that
(c·Δt/Δτ)² - (Δx/Δτ)² = c²
A priori, we could certainly imagine this relation not to hold!
However, once we've baked this relation into the geometry of spacetime, we can of course take the more abstract perspective described above and think about reparametrization-invariant dynamics, with choice of eigentime as parameter an insignificant way to fix an arbitrary gauge.
I see. Good point, it depends on how you define four velocity and how exactly you interpret "moving through time with the speed of light". I personally think that "we move through time at the speed of light" is a very confusing way to explain that physical fact about how clocks tick. I'd explain the same fact as follows:
If you have a clock moving through spacetime along some path p(r), put tickmarks at regular intervals [1] of its arc length ∫|p'(r)|dr. Those tickmarks indicate when the clock ticks.
[1] e.g. choosing the length of an interval by matching it to a one second tick of a stationary reference clock.
I spent quite some time being confused about this.
In differential geometry, we traditionally parametrize curves by a parameter "t" and think of it as "time", so the parametrization allows us to "walk along" a curve. This is very intuitive geometrically of course.
But in relativity, we also have a "time" coordinate (or at least a timelike unit-length tangent vectors), which then completely oposes this geometric intuition about curves.
Of course now physicists decide to rename well-established concepts and start calling "arc length parametrization" by "proper time parametrization", which makes it sound like it is something special, while it, as far as I can tell, has no actual physical meaning.
> vectors in a context where they certainly can be of varying magnitudes
Some vectors can vary in magnitude with time, yes. But rotating your coordinate system does not change the magnitude of any vector in Euclidean space. A Lorentz transformation is the spacetime analogue of rotating your coordinate system, and similarly does not change the magnitude of any vector.
