Translation:Having finished the work, he went home.
It's fininte because it does actually refer to iris. Fininte la laboron is working as an adverbial phrase modifying the whole sentence here. It provides time information for iris, just like "post kiam li finis la laboron" or "hieraŭ".
It could not be fininta because an -a form appears in only two syntactical situations:
(A) as an attributive adjective before or after an -o word. (Eg. la granda viro = the big man; la fininta viro = the man who has finished)
(B) in a predicate, generally with a form of 'esti', 'iĝi', 'resti' or a similar verb, or be in a kind of secondary predicate like Mi farbis la ruĝan pordon nigra.
For this same reason, we can say rapide li iris hejmen but not
rapida li iris hejmen. Yes, in a way, "rapid-" describes li, kind of, but it actually describes how the sentence is carried out and syntactically modifies the finite verb, just like the "fininite" above describes when the sentence is carried out.
Perhaps it's easier to understand if you compare it with a present continuous participle since other European languages often only have that.
"Ŝi eniris la ĉambron parolante laŭte" - She came into the room talking loudly - Sie kam laut redend ins Zimmer - Elle venait dans la pièce en parlant à haute voix.
You do not say "parlante" in French even though she is feminine.
The participle here acts like an adverb. It describes that the coming and the talking is simultaneous.
Esperanto also has this kind of participle in the past, but it's still adverbial.
English can say "Having finished the work", but in German (for example), "die Arbeit beendet habend" would sound strange. But "Fininte la laboron" is fine in Esperanto.
If you want to use it as an adjective, to say that he is in a state of having finished, that is possible: "Li estas fininta la laboron."
But in this sentence, it's an adverb, "having finished".
It has always surprised me that in Spanish (with which I'm more familiar than with French) you'd say 'ella salió cantando' (she left [while] singing), not 'cantanda' nor 'cantandamente'. But then, no Spanish-speaker would say this is an adjective or an adverb, just a form of the verb. In Esperanto, it troubles me that the participle kantant- can behave as an adjective (la kantanta virino), as a more 'verbish' adjective (la virino estas kantanta) and as an adverb (kantante la virino eniris). (Not to mention kantanto, and I wouldn't be surprised to find 'sxi kantantas'.) In spite of your kind and helpful explanation, the adverb doesn't sit easily with my linguistic instincts.
This is one thing that makes Esperanto more complicated than it seems at first.
The endings show you what part of speech a word is being used as, but there's no way to know from just looking at a root, what its "original" part of speech is.
OK, it may seem intuitive that knabin- is a noun root, and knowing that noun-roots can't be used as verbs to express "be + noun" is one thing, but there is no way to know from mi martelis la najlojn en la lignon and mi kombis mian hararon that martel- is a noun root (martelo 'a hammer') and komb- is a verb root (kombi 'to comb').
The tools for each action are:
martel/o = a hammer komb/il/o = a comb
The name for the action itself is
martel/ad/o = hammering komb/o = combing
Another example, apparently gej- is a noun root (gejo), not an adjective root (geja), so you can say li estas samseksema or shorter, li samseksemas but li estas geja cannot, at least in theory, be shortened to li gejas, and there's no way to know this without checking a dictionary for the original part of speech ... of everything.