I mark the moment the paper is on the carpet — picked up quietly (4-5)

'i mark the moment' is the definition.
(a time-stamp records a particular moment or time)

'paper is on the carpet picked up quietly' is the wordplay.
'paper' becomes 'times' (The Times newspaper).
'is on' means one lot of letters go next to another (in a down clue, letters appear on others).
'the carpet' becomes 'mat' (a mat is a floor covering).
'picked up' shows that the letters should be reversed in order.
'quietly' becomes 'p' (musical symbol).
'mat' backwards is 'tam'.
'times'+'tam'+'p'='TIME-STAMP'

'the' is the link.