Bird that's right? Stick around! (5)
I believe the answer is:
crane
'bird' is the definition.
(crane is a kind of bird)
'right? stick around' is the wordplay.
'right?' becomes 'r' (common abbreviation).
'stick' becomes 'cane' (cane is a kind of stick).
'around' indicates putting letters inside.
'r' put within 'cane' is 'CRANE'.
'that's' is the link.
(Other definitions for crane that I've seen before include "Lifting machine or long-legged bird" , "Device for picking up" , "Wading bird - lifting device" , "American writer" , "Bird; dockside machine" .)