Having a lead, stops 1,000 dogs, say, getting nits (7)

'nits' is the definition.
(both can mean fools)

'having a lead stops 1000 dogs say' is the wordplay.
'having a lead' becomes 'up' (eg being up in a football match).
'stops' is an insertion indicator (stopping can mean blocking a gap).
'1000' becomes 'M' (Roman numeral).
'dogs say' becomes 'pets' (dogs are a type of pet).
'm'+'pets'='mpets'
'up' going into 'mpets' is 'MUPPETS'.

'getting' is the link.