Translation:He will talk to me as soon as he finishes eating.
The verb tense of "finire" is incorrect in this Italian sentence. As "finire mangiare" is an action which takes place before the future action "parlare", "finire" should use the "futuro anteriore" tense, not "presente": Mi parlerà appena avrà finito de mangiare. Please report it.
It should be "avrà finito". Reference: https://italian.stackexchange.com/questions/10976/future-dependent-actions-verb-tense