Stick around, admitting love by the end of fiery speech (6)
I believe the answer is:
eulogy
'speech' is the definition.
(speech at a funeral)
I cannot understand how the remainder of the clue works.
(Other definitions for eulogy that I've seen before include "Funeral speech" , "parting words?" , "Expression of praise for someone recently dead" , "Formally expressed praise" , "'Encomium, paean (6)'" .)