The Trim Open Paths operation is used to trim off one or multiple selected open paths where they are intersected by other visible, unlocked paths (or by self-intersections). By default, the trimming can occur from either end of the path (the shorter end, if the path is intersected only once; otherwise both ends), but Shift can be held while clicking the button to force trimming from only one end; which end depends on the preferences setting.
QuickOps Trim Open Paths
Clicking the icon to the right of the button will bring up the Trim Open Paths Preferences dialog:
QuickOps Trim Open Paths Preferences
Specifies which side of the path to trim if the button is clicked with Shift held down.
When enabled, and the trimmed path has a variable width stroke, the stroke profile will be split to keep the width of the untrimmed section as close to the original as possible.
QuickOps Trim Open Paths - Split Variable Width Profiles