This is Definition 10 from Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neill, page 7:
On both sides of Definition 10, he is differentiating with respect to functions, not independent variables. What does that mean?
He defines both xi and ui to be functions, not coordinates.
He does so on the first two pages of the book:
Here's a picture of what the functions do:
Let's start with multivariable calculus. On the plane, you should have no qualms with the fact that the coordinates $x, y$ are functions $\mathbb R^2 \to \mathbb R$. If you were to take the function $x$ alone, it doesn't make sense to talk about a partial derivative $\partial/\partial x;$ but when you have $x$ and $y$ together, you have a complete coordinate system, and thus you can define $\partial/\partial x$ by holding $y$ constant.
The situation on manifolds is the same - once you have fixed a coordinate system, you can define corresponding partial derivatives. Definition 10 is doing exactly this - for a coordinate system $x^i,$ we are defining the corresponding partial derivative operators. Note that this definition does not allow you to obtain e.g. $\partial/\partial x^1$ from the function $x^1$ alone - you need to know more about $\xi.$
Remember that the $\partial/\partial x^i$ on the left side of the equation is what is being defined - so to interpret it rigorously we just need to make sense of the right side. On the right side we have derivatives with respect to the standard coordinates $u^i$ on $\mathbb R^n$ - these are not some exotic "derivatives with respect to functions", but just the usual partial derivatives in $\mathbb R^n,$ like the $\partial/\partial x$ I discussed earlier.