Let's dissect this phrase:
"his delight evident"
his - the possessive pronoun indicates the individual who is referenced (pretty straightforward)
delight - the noun possessed by the individual
evident - an adjective describing the noun
Essentially, the author has flipped the usual adjective-noun clause, which would have originally read "his evident delight" to a noun-adjective clause, thereby emphasizing the adjective, instead of the noun.
To elaborate, look at the following sentences:
"She gave him a gift, to his evident delight."
In this sentence, the delight is the emphasis. The delight is being characterized by the word "evident," but "evident" is merely a descriptor.
"He thanked her profusely, his delight evident."
In this sentence, the fact that the delight was obvious to those around him is the emphasis. While "evident" is still the descriptor of the delight, the fact that his delight was evident is more important to understand than that the individual was delighted in the first place.
Does this help?