Syntactically speaking, his lesson is parenthetical, or nonrestrictive, information. Analyzing the sentence, it would be possible to drop it:
a father's gnarled disease
, his lesson, from his desk
So, while it may be understood that he is reciting his lesson, what the sentence is saying at its core is that he is reciting the gnarled disease. While they are referring to the same thing, its the disease itself that has the primary meaning in the sentence.
In this case, reciting is being used figuratively. Or, at the very least, the recitation is not in the traditional verbal sense but in a physical sense.
In other words, he speaks the disease on the frame of his body—in the same way that we normally speak words, or thoughts, with our mouths.
That was just a syntactical analysis. It doesn't mean that's how people necessarily interpret it. Some interpretations don't follow from literal syntax.
For the literal syntax to match the interpretation suggested by those online sources, it could be rephrased in the following way:
He had a disease. His disease was his lesson. He recited his lesson from his desk.
It's still figurative (and that part of what I said still applies), but it no longer makes use of anything parenthetical.
However, it also completely changes the style and mood of the writing.