On building a nest, start to feel secure again (8)

'secure again' is the definition.
('fasten' can be a synonym for 'secure' so perhaps 'refasten' is OK for 'secure again')

'on building a nest start to feel' is the wordplay.
'on' becomes 're' ('on' can be mean 'with reference to').
'building' is an anagram indicator.
'start to' suggests taking the first letters.
The initial letter of 'feel' is 'f'.
'a'+'nest'+'f'='anestf'
'anestf' with letters rearranged gives 'fasten'.
're'+'fasten'='REFASTEN'