the tips has a limited explanation. but for a general answer it is enough. among its uses is to express a wish or give an order. lascia is the imperative and is followed by the subjunctive describing what you must allow. subjunctive isn't really a tense in italian. it's a mood that describes a state of being.
Buongiono! This Italian sentence can also be translated: "Let him finish" or "let her finish." (Duo still does not accept those legitimate alternative translations. Mine was just rejected. Duo insisted on "Let me finish".) The Italian conjugated form "finisca" can refer to the first person singular, the second person singular and the third person singular. Since there are no specific pronouns to clarify the matter, all these possibilities should be accepted in the English translation.)
"me" (or also him / her) is part of "finisca", as the subjunctive has a different ending than the indicative.
See congiuntivo - presente at this link: http://www.italian-verbs.com/italian-verbs/conjugation.php?verbo=finire