Let $D$ be the discrete space of cardinality $\kappa$, and let $x_0$ be a point not in $D$. Let $A=D\cup\{x_0\}$ be the one-point compactification of $D$, with $x_0$ as the added point. The compact subsets of $D$ are precisely the finite sets, so open nbhds of $x_0$ in $A$ have the form $\{x_0\}\cup(D\setminus F)$ for finite $F\subseteq D$. Let $\mathcal{O}$ be the topology on $A$; what, exactly, is $\mathcal{O}$?
Let $S\subseteq A$. If $x_0\notin S$, then $S\subseteq D$, so $S\in\mathcal O$ iff $S$ is open in $D$. But $D$ is discrete, so $S$ is open in $D$ $-$ all subsets of $D$ are open in $D$ $-$ and therefore $S\in\mathcal{O}$. If $x_0\in S$, then $S$ must contain an open nbhd of $x_0$, so there must be a finite $F\subseteq D$ such that $\{x_0\}\cup(D\setminus F)\subseteq S$. But then $A\setminus S\subseteq A\setminus\Big(\{x_0\}\cup(D\setminus F)\Big)=D\setminus(D\setminus F)=F\;,$ so $A\setminus S$ is finite. In other words, $S\in\mathcal O$ iff either $x_0\notin S$, or $x_0\in S$ and $A\setminus S$ is finite. Finally, this is the same as saying that $S\in\mathcal O$ iff $x_0\notin S$, or $A\setminus S$ is finite, which is exactly the definition of the topology $\mathcal O$ given in Example 1.1.8. Thus, $A(\kappa)$ is, as azarel said, the one-point compactification of the discrete space of cardinality $\kappa$.
$A(\omega)$ is indeed homeomorphic to $\omega+1$, a simple sequence with a limit point. If $\kappa>\omega$, it’s similar, but it’s not a sequence: it’s a set of $\kappa$ isolated points and one limit point, $x_0$, with the property that every open nbhd of $x_0$ contains all but finitely many of the isolated points. This means, among other things, that if $S$ is an infinite set of isolated points, then $x_0\in\operatorname{cl}S$: every open nbhd of $x_0$ must contain all but finitely many points of $S$. It also means that $A(\kappa)$ is compact: once you’ve covered the point $x_0$, there are at most finitely many points left to be covered. (Of course this is also clear from the fact that $A(\kappa)$ is a one-point compactification!)