Understanding the definition of a set with $C^k$ boundary and of the outward pointing normal vector field

The definition says, geometrically, that the boundary of $U$ is locally the graph of a function of class $C^{k}$, i.e., a sufficiently small piece of $U$ near a boundary point $x_{0}$ is (possibly after re-indexing coordinates) the super-level set $$ F(x_{1}, \dots, x_{n-1}, x_{n}) := x_{n} - \gamma(x_{1}, \dots, x_{n-1}) > 0. $$

For example, the function $\gamma(x) = |x|x^{k}$ is of class $C^{k}$ but not of class $C^{k+1}$, so the (unbounded) region $$ U = \{(x, y) \text{ in } \mathbf{R}^{2} : y - \gamma(x) > 0\} $$ has boundary of class $C^{k}$, but not of class $C^{k+1}$. (At most points the boundary is real-analytic, but this example should convey the idea. If you have "more interesting" examples of $C^{k}$ functions, you can cook up correspondingly more interesting regions.)

If $\partial U$ is of class $C^{1}$, then by definition a small piece of $\partial U$ may be written as the graph of a $C^{1}$ function. Consequently, at each boundary point $x_{0}$, there is a well-defined tangent space $T_{x_{0}}$, of dimension $(n - 1)$; the outward unit normal $\nu(x_{0})$ is the unique unit vector orthogonal to $T_{x_{0}}$ and satisfying the directional derivative condition $\nu(F) < 0$, with $F$ a local defining function for the boundary, as above. Geometrically, $\nu(x_{0})$ spans the orthogonal complement of $T_{x_{0}}$ and points toward the exterior of $U$, where $F < 0$.