Yet candor requires us to acknowledge that of all the men collected in the Roman camp he was the fittest to have extricated the army from its embarrassments, and have conducted it, without serious disaster or loss of honor, into a position of safety.
The sentence means that the man he is speaking of was the best choice (the fittest)
to get the army out of its difficulties (to extricate it from its embarrassments)
and to lead (to conduct) it to safety.
The to-infinitive in that use is an infinitive of purpose. In the present, we could say
He is the best man to save the army.
meaning that he is the best man for that purpose.
In speaking of the past, we could say
He was the best man to save the army.
That statement simply sets the first sentence in the past tense.
The writer of this text is setting the same idea in the past, but also with a perfect infinitive.
... he was the fittest to have extricated the army ...
That is because he is writing about an unreal situation in the past. While the man they chose (Julian) was the best man for the purpose, he was killed in battle, and so he did not save the army.
About the perfect infinitive for unreal past situations:
englishgrammar.org perfect infinitve
"Unreal past situations
The perfect infinitive is often used after verbs like mean, be, would like etc., to talk about unreal past situations.
She was to have returned yesterday, but she fell ill.
I meant to have posted the letter, but I forgot. (I did not post the letter.)"
(While these examples aren't infinitives of purpose, they show the use of the perfect infinitive for past unreal situations.)