The context is explained by the previous sentence, which "tells" what the specific description of a painting "shows":
They leaned towards the honest and explicit in art, a picture, for instance, that told its own story, with generous assistance from its title.
This says that their taste in paintings is to prefer things that don't need much explanation: a picture that tells its own story, the story being clear enough just from looking at the picture itself and its title or caption.
A riderless warhorse with harness in obvious disarray, staggering into a courtyard full of pale swooning women, and marginally noted "Bad News," suggested to their minds a distinct interpretation of some military catastrophe.
This description is of a painting whose context and "story" is clear just from looking at it and its "marginal note" or caption. A warhorse would suggest some military connection; the horse being in disarray, staggering, and without its rider, suggests that some ill has befallen it; and the women being pale and swooning sugests that its very appearance heralds some bigger misfortune. All of this taken together points towards a "military catastrophe" such as a grave defeat in battle, this horse being a survivor even if its rider might be dead or injured.
So the picture tells its own story, and just in case it's not clear enough just from looking, the caption lends some "generous assistance". Knowing that the painting depicts "Bad News" just confirms what could have been guessed from looking at it: the horse's appearance heralds some bad news, presumably something related to its experience as a warhorse.
They could see what it was meant to convey, and explain it to friends of duller intelligence.
This sentence again tells us what the second-to-last one did: what they like in paintings is to be able to see what it's meant to convey. Their inner art critic is good enough to do that, and the implication is that they may enjoy feeling superior to their friends who have less experience of interpreting art, referred to somewhat superciliously as people "of duller intelligence".