In Chapter 1, "Vector Analysis" of Griffith's Electrodynamics he says at some point
a vector is any set of three components that transforms in the same manner as a displacement when you change coordinates. As always, displacement is the model for the behavior of vectors$^3$
And the footnote
If you're a mathematician you might want to contemplate generalized vector spaces in which "axes" have nothing to do with direction and the basis vectors are no longer $\hat{x},\hat{y}$ and $\hat{z}$ (ineed there may be more than three dimensions). This is the subject of linear algebra. But for our purposes all vectors live in ordinary 3-space (or, in Chapter 12, in 4-dimensional space-time).
My question is about how to understand especially the first quote.
What does it mean for components to transform in the same manner as displacement?
Displacement seems to mean the change in a position vector, and this change is itself a vector. We can do vector operations on a displacement vector. Now, I'm guessing it is the case that when we change the coordinates (technically, I think this means that we are changing the basis in the same vector space, though I am not sure) the displacement vector (all vectors?) stays the same in terms of magnitude (but not necessarily direction?).
My question is how to put these pieces together into a coherent picture that explains Griffith's first quote above.