Does this English translation actually capture how this sentence differs from "Li legis la klarigon" or "Li estis leganta la klarigon"? I put "He is one who has read the explanation," which though stilted, seems to me to express what is going on with this verb.
Yes, I think it's probably the best English translation.
Li legis: he read.
Li estas leginta: he has read.
Li estis leganta: he was reading.
Of course, "li legis" could also mean "he has read" or "he was reading"; using the participles lets you be more specific but is not required.
That is the impression I get as well. Of course, we would rarely say "He is one who has read the explanation," but it seems that this participial construction in Esperanto would be pretty rare as well.
Maybe it's just me but I use them a lot, even if it makes me an obvious anglaparolulo.
A, mi aŭdis pri tiu manlibro.
Sed mi volas diri, ke "la manlibro, ni ĉiam pri aŭdas" ne estas gramatike ĝusta. "Is it the manual? We always about hear."
English can leave out relative pronouns as in "The manual (which) we always hear about", but Esperanto can't - you need an explicit "kiu", in this case. "The book he read" is not "La libro li legis" but "La libro kiun li legis".
And "pri" has to stand before a noun or pronoun; it can't be at the end by itself, much less in front of the verb. So "la manlibro, pri kiu ni ĉiam aŭdas".
How come "He has read the explanation" and "He had read the explanation" are different and one results in a correct answer while the other doesn't?
Had read would be the pluperfect or past perfect or anterior perfect. You should use that when talking about something that occurred before something else that you are describing in the past tense.