On page 102 of his book Strongly Elliptic Systems and Boundary Integral Equations, McLean defines the trace operator $\gamma:\mathcal{D}(\bar{\Omega})\rightarrow\mathcal{D}(\Gamma)$ as $$\gamma u=u|_\Gamma,$$ where $\Omega$ is a Lipschitz domain, $\Gamma$ the boundary of $\Omega$, i.e. $\Gamma=\partial\Omega$, and $$D(\bar{\Omega})=\{u:u=U|_\Omega\ \text{for some}\ U\in\mathcal{D}(\mathbb{R}^n)\}.$$ The definition of $D(\mathbb{R})$ follows from the definitions below, where $\Omega$ is an open subset of $\mathbb{R}^n$ (doesn't need to be a Lipschitz domain). $$\mathcal{D}(\Omega)=C_{\rm{comp}}^\infty(\Omega)=\{u:u\in C^\infty_K(\Omega)\ \text{for some}\ K\Subset\Omega\},$$ where $$C_K^\infty(\Omega)=\bigcap_{r\ge0}C_K^r(\Omega),$$ where $$C_K^r(\Omega)=\{u\in C^r(\Omega):\rm{supp}\,u\subseteq K\},$$ where $$C^r(\Omega)=\{u:\partial^\alpha u\ \text{exists and is continuous on}\ \Omega\ \text{for}\|\alpha|\le r\}.$$
I have a rather weak Mathematics background, and so I might confuse some things here, but the way I understand the restriction $|_\Gamma$ is that it effectively diminishes the set of elements that can be given to whatever functions it is applied, to elements in $\Omega$. That being said, I find it rather confusing that the operator $\gamma$ takes elements from $\mathcal{D}(\bar{\Omega})$ (which are restricted to $\Omega$) and maps them to functions defined on $\Gamma$. After all, $\Gamma\cap\Omega=\emptyset$ by definition.
Here is the reason, why I want to know about this in the first place. Let $\Omega$ be a bounded connected Lipschitz domain in $\mathbb{R}^2$ and let $u$ be such that it is $1$ in the closure of $\Omega$ in $\mathbb{R}^2$ and zero otherwise. I am now interested in the expression $$\gamma^+u|_{\bar{\Omega}^c}-\gamma^-u|_\Omega$$ where $\gamma^+$ and $\gamma^-$ are the trace operators for $\bar{\Omega}^c$ and $\Omega$ respectively.