What the State is, and Where it comes from, when Generating Reachability Graphs
Timed automaton case: Here, a state is typically composed of the current location and the values of all clock variables. Thus, timed automata induce transition systems with an infinite number of states.

The usual approach in this context is thus to build a finite representation of this state set, which is a finite transition system. In the paper "An Efficient State Space Generation for Analysis of Real-time Systems" by Inhye Kang and Insup Lee that you linked to, the state set for this finite transition system is given on page 5, left column. The classical approach involves building a so-called region graph, which is referenced in column 2 of page 4.

Note that CTL model checking of the resulting transition system makes little sense, as the "next-time" operator of CTL does not have a usable semantic in this case. Rather, there are specialized logics (such as TCTL) to reason about properties of timed systems.

Petri net case: In the Petri net case, the states are the possible markings of the network. Note that the state set could be finite or infinite depending on the network. In paper that you referenced ("Petri Net Reachability Graphs: Decidability Status of First Order Properties" by Philippe Darondeau, Stéphane Demri, Roland Meyer, Christophe Morvan), the reachability graph is defined on page 4 as consisting of all markings and having a transition between a marking $M$ and $M'$ if we have $M[t\rangle M′$ some $t$. This notation is in turn explained at the beginning of the page, in Definition 2.