In somewhat of a similar fashion to Drag Notes, they are slid/flicked along the line shown on the note.
Shown here is the Cytus Omega tutorial for Flick Notes:
Since Cytus Omega has yet to release, there is currently little information about where Flick Notes are used.
Unlike Cytus Omega, in Cytus II you flick these in any direction.
Shown here is the Cytus II tutorial on flick notes: