Space doesn't expand. The standard coordinates (FLRW) do.
The way FLRW coordinates cover spacetime is similar to the way polar coordinates cover Earth's surface. If you look at a top-down map of a region of Earth away from the equator, like this one:
you'll see that the lines of constant latitude are curved, and the lines of constant longitude are roughly straight but not parallel. The same thing happens to FLRW coordinates: surfaces of constant cosmological time are curved (extrinsically in the full spacetime) and lines of constant cosmological position are straight (they're geodesics) but they aren't parallel.
If two people at the same latitude but different longitudes in Australia both walk due north, they will get further apart. This is not because Australia expands as you go north, but because their paths aren't parallel. They are literally walking away from each other. The same is true in spacetime.
On the other hand, if you move an east-west pointing ruler north, its length in longitudinal degrees changes, but its length in meters doesn't. Physically, all you've done is move an object of constant length from one place to another. Polar coordinates are great for some purposes, but if you're only interested in the local properties of rulers, they're outright counterproductive: they make "effects" appear to exist that actually don't.
Unfortunately there are no light cones in this otherwise accurate analogy. But the local geometry of spacetime is everywhere the same, like the Earth is locally flat if you look at a small enough piece of it, and the geometry being the same everywhere means in particular that you'll always measure the same speed of light.
