Regardless of where you got it, I think what we have here is a badly constructed English language proficiency test. Both from the grammar and sentence structure standpoints, (D) “and inspired thousands of others to attempt to overcome” simply couldn’t be the correct answer.
See what happens when we use answer D in that sentence:
“With his sub-four minute mile Bannister broke a psychological barrier, and inspired thousands of others to attempt to overcome seemingly insurmountable hurdles.”
The comma preceding the additive clause “and inspired thousands of others to attempt to overcome seemingly insurmountable hurdles” makes the sentence structurally defective. Since both the first clause “Bannister broke a psychological barrier” and “(Bannister) inspired thousands of others to attempt to overcome seemingly insurmountable hurdles” have the same subject, that comma is unnecessary and only serves to truncate the compound sentence. The correct construction is to treat the verb phrases “broke a psychological barrier” and “inspired thousands of others…” as a simple compound verb phrase unpunctuated by a comma (it would also be in the interest of clarity to put a comma after the prepositional phrase “with his sub-four minute mile”), as follows :
“With his sub-four minute mile, Bannister broke a psychological barrier and inspired thousands of others to attempt to overcome seemingly insurmountable hurdles.”
A comma after the first coordinate clause would have been necessary had the second verb phrase used the pronoun “he” for “Bannister” as its subject, in which case it would have become a full-bodied second coordinate clause on its own, as follows:
“With his sub-four minute mile Bannister broke a psychological barrier, and he inspired thousands of others to attempt to overcome seemingly insurmountable hurdles.”
For largely the same reason, answer (E) “and inspired thousands of others to attempt overcoming” is also an incorrect answer.
(C) “inspiring thousands of others to overcome” is actually a possible answer purely from a grammatical and structural standpoint. From a semantic standpoint, however, it’s incorrect because it gives the wrong sense that thousands of other people have already overcome insurmountable hurdles by following Bannister’s example. This, of course, isn’t the case at all; their surmounting of their individual hurdles is just an expectation, not yet a reality.
This leaves either (A) “inspiring thousands of others to attempt overcoming” or (B) “inspiring thousands of others to attempt to overcome” as the only possible correct answers. Both the form “to attempt overcoming” (infinitive + gerund) and the form “to attempt to overcome” (infinitive + infinitive) yield a semantically valid sense for the sentence, so they are both correct answers. Stylistically and euphonically, however, I prefer answer B, in which case the sentence should read as follows:
“With his sub-four minute mile Bannister broke a psychological barrier, inspiring thousands of others to attempt to overcome seemingly insurmountable hurdles.”
I must say that the explanation you cited justifying (D) “and inspired thousands of others to attempt to overcome” as the correct answer is pure nonsense. There is neither logical nor imperative need for the verbs “inspired” and “broke” to be set in parallel in that sentence, and to say that the form “attempt to overcome” is correct because it’s idiomatic is a vapid, insupportable explanation. It’s evident that the developer of that test sentence isn’t very conversant with the English language, in the process providing faulty answer choices in many respects.
To avoid befuddling our English, we should guard against such low-quality English proficiency tests that litter the World Wide Web.