When the recording is done, please play it back for me.
Literal
Recording [subject-が] finished [conditional-たら] it [object-を] play-back [and-て] let-hear [request-てください].
~たら is a temporal conditional ('when X is done'). 聞かせてください uses the causative 聞かせる ('to let hear, play for someone') wrapped in a polite request — literally 'cause me to hear,' which functions as 'play it for me, please.'