Introductory material on splines I am looking for a basic, step-by-step introduction into modelling with splines.
(I have encountered splines while teaching another topic. The textbook I am using does not cover splines in sufficient detail for the students to be able to replicate things by themselves. I hope there is some good introductory material out there so that I do not have to reinvent the wheel and write my own notes on the topic.)

Echoing my comment, I would put forth the monograph "Nonparametric Regression and Generalized Linear Models" by Green, Silverman. It is not highly meticulous as the likes of de Boor's "A Practical Guide to Splines" (after all, it's not spline-centric treatise) but it is comprehensive and provides a lucid introductory account of splines: the authors motivate the concept of spline by observing that when a spline is bent in the shape of a curve $g, $ the leading term in the strain energy is $\propto \int {g^{\prime\prime}}^2.$ In doing that, they basically "quantify" the roughness of a curve.
Now, that is enough of an intuitive opening to a new realm. Again, this book doesn't delve too much in the functional analysis formalism as in "Smoothing Splines: Methods and Applications" by Wang but this must not deter any one to set it aside. The authors cover interpolating, cubic, natural cubic, smoothing splines, their properties, constructions, plotting, existence of minimizing spline and associated algorithms. There is a chapter on partial spline (unfortunately, I didn't cover that, so won't comment).
In all, while this is definitely not a spline centric book, it's worth a try. I am not aware of OP's students' level but as a student myself, I enjoyed the first reading with enough relevant mathematical materials for a first read.

Recommendation:
Nonparametric Regression and Generalized Linear Models: A roughness penalty approach, P. J. Green, B. W. Silverman, Chapman & Hall, $1994.$
