The close correlation between cooling flows and emission-line nebulae in clusters of galaxies has been recognized for over a decade and a half, but the physical reason for this connection remains unclear. Here we present deep optical spectra of the nebula in Abell 2597, one of the nearest strong cooling-flow clusters. These spectra reveal the density, temperature, and metal abundances of the line-emitting gas. The abundances are roughly half-solar, and dust produces an extinction of at least a magnitude in V. The absence of [O III] 4363 emission rules out shocks as a major ionizing mechanism, and the weakness of He II 4686 rules out a hard ionizing source, such as an active galactic nucleus or cooling intracluster gas. Hot stars are therefore the best candidate for producing the ionization. However, even the hottest O stars cannot power a nebula as hot as the one we see. Some other nonionizing source of heat appears to contribute a comparable amount of power. We show that the energy flux from a confining medium can become important when the ionization level of a nebula drops to the low levels seen in cooling-flow nebulae. We suggest that this kind of phenomenon, in which energy fluxes from the surrounding medium augment photoelectric heating, might be the common feature underlying the diverse group of objects classified as LINERS. ; Comment: 33 Latex pages, including 16 Postscript figures, to appear in 1997 September 1 Astrophysical Journal