After falsehood mean to keep out of sight (3,3)
I believe the answer is:
lie low
'keep out of sight' is the definition.
(I've seen this before)
'after falsehood mean' is the wordplay.
'after' means one lot of letters go next to another.
'falsehood' becomes 'lie' (lie is a kind of falsehood).
'mean' becomes 'low' (both can mean nasty or dishonourable).
'lie'+'low' is 'LIE LOW'.
'to' is the link.
(Other definitions for lie low that I've seen before include "Stay quiet and/or hidden" , "Conceal oneself or one's intentions" , "Use bottom bunk, perhaps" , "Stay out of sight, hide horizontally?" , "'Keep out of sight, avoid attention (3,3)'" .)