Edit: To avoid confusion, the authors aren’t referring to thermodynamic reversibility (i.e., zero entropy generation). This never occurs in real life. They’re referring to whether a strongly exothermic reaction can be run in reverse.
Broadly, Nature prefers both strong bonding (low enthalpy $H$) and many possibilities (high entropy $S$). (This is why we see spontaneous minimization of the Gibbs free energy $G = H-TS$.)
Consider the chemical reaction
$$A+B\to AB,$$
where AB has an extraordinarily low enthalpy. I mean, really great bonding there. Some would describe this reaction as occurring completely—as being irreversible. If you have A and B around, putting aside any kinetic limitations, it's inevitable that they're going to permanently form AB, they might say.
But this isn't truly the case. In a universe with AB only, regardless of how strongly it's bonded, the entropy increase from a single dissociation of an AB molecule into A and B would increase total entropy enormously, as the A and B components could uniquely occupy any location in the cosmos. This entropy increase provides a sufficient driving force for some degree of dissociation (or reaction reversal).
The dissociation tendency increases with increasing temperature; on the molecular level, you could interpret this as more surrounding kinetic energy tending to break the AB bond by chance, or on the system level, you could interpret this as temperature mediating the strength of the $TS$ term above.
The upshot then is that all reactions are reversible, in the sense that we could reverse them by removing enough of the reactants. (This is one implication of Le Chatelier's principle.)