I don't understand what is going on here. What is $U$ and $\overline U$ , why do we need to have $C^{2}(U)\cap C(\overline U)$ ? What is the partial derivative $U$ mean ? What is "$U$ is connected "?Overall I don't understand almost everything . Please help.
On
$U$ is a domain, an open subset of $\mathbb{R}^n$, presumably bounded (though not necessarily for the strong maximum principle); $\bar{U}$ denotes its closure; $\partial U$ denotes its boundary, $\partial U := \overline{U}\cap \overline{U^c}$. A set $U$ is connected if it cannot be written as the disjoint union of two non-empty open sets. We require $U$ to be continuous up to the boundary because $u \in C^2(U)$ doesn't imply $u$ is even defined on the boundary -- a simple example is $u(x) = 1/x$ on $(0,1)$ -- $u \in C^\infty((0,1))$. We also only know one-sided derivatives of $u$ at the boundary.
Most of this notation is standard, can be found in the appendix of Evans' book, and if it is confusing to you, you should certainly start with some foundations of analysis.
With the notation clarified, the theorem tells us that under the hypothesis $u$ attains its maximum on the boundary of its domain. Furthemore, if the domain is connected and if there is a point inside the domain where this maximum is also reached, then necessarily $u$ is constant function.