Insect starts to chirp: "I'm cricket, and drummer also!" (6)
I believe the answer is:
cicada
'insect' is the definition.
(cicada is a kind of insect)
'starts to chirp i'm cricket and drummer also' is the wordplay.
'starts to' says to take the initial letters.
The first letters of 'chirp im cricket and drummer also' is 'CICADA'.
(Other definitions for cicada that I've seen before include "Shrill bug" , "Noisy tropical insect" , "Flier" , "Stridulating insect" , "six-footer?" .)