Obvious Dominance and Milipede Games (Question to Pycia and Troyan (2019)
It is a convoluted definition because condition 3: 

"At all $h$, if there exists a previously unclinchable payoff x that becomes impossible for agent $i_h$ at $h$, then $C_i^\subset (h) \subseteq C_i^h (h)$." 

means that at every history if there is a payoff that player $i_h$ could not have secured (or clinched), but was feasible at every previous history where some other player was taking decisions and now is not feasible, then the set of clinchable payoffs cannot decrease. 

That is, the set of clinchable payoffs is not allowed to decrease, except when some special condition is met. 

Figure 1 is an example of a millipede game because the set of clinchable payoffs never decreases for players $j$ and $k$. It only decreases for player $i$ when (it goes from $\{x,y,z\}$ to $\{x,y\}$ (from the second to the third move), but this is fine because in player $i$'s third move the only payoff that was previously unclinchable is $w$, but it has not become impossible at that history (since the only impossible payoff is $z$ that was taken by player $j$).

Hope this helps.