James being very Jamesian: long sentences; obscure and perhaps thin emotional stakes; focus on artistic personalities and their tribulations; and a dollop of homosexual tension.
Dencombe is not elderly, but he's sick, and he's seemingly given up hope that he'll accomplish anything else artistic with his life. So when a young doctor praises Dencombe's most recent book--before realizing that he's talking to Dencombe--it's a shot in the arm for his emotional well-being. Unfortunately for the young Doctor Hugh, it's a potentially terminal condition for his hopes of inheriting from the wealthy and whimsical old woman who he's caring for. Which puts Dencombe in a tight spot: he wants his young friend to succeed, but he also just wants his young friend.
That's the plot; and hopefully I've made it clear why I say the stakes are pretty thin. I mean, the protagonist has so little agency here that his main decision is basically just to figure out how he feels about things that are going on around him. The story wouldn't be materially different if Dencombe spent most of the story unconscious.
Of course, the plot isn't the only thing going on here; most of the action and narrative focus on Dencombe's feelings, from the sad sack opening to the less-sad sack ending. All of which gives the story a certain narcissistic--and autobiographical--feeling. As others have noted, Dencombe's mid-life crisis maps pretty well onto James's own mid-life situation.
Which makes it extra funny when Dencombe finds himself swept away by his own book, as he "surrender[s] to his talent"; and then realizes that even sensitive and admiring Doctor Hugh just doesn't get him. Since we spend so much time in his mind, the story seems to lean towards Dencombe's POV--as if we're supposed to feel bad for him, and by extension, for James.
But that's the ungenerous and un-fun reading. For fun reading, we could look at the vaguely homo-erotic relationship between Dencombe and Hugh. Young Hugh may be torn between a dying old woman--who has nothing to recommend her but her money--and a dying old man--who has nothing to recommend him but his AWESOME talent--but it hardly seems like a fair fight. When Hugh notes, "I don't get on with silly women," it sounds a lot more like he doesn't get along with women at all. (After all, he's not interested in either the old woman or her younger traveling companion.) And when Hugh praises Dencombe, well:
"You're a great success!" said Doctor Hugh, putting into his young voice the ring of a marriage-bell.Of all the bells in all the world, James would choose a "marriage-bell."