Use the Pen Tool to add an anchor point to the path where you want to cut it.
Then use the Direct Selection Tool to select the end anchor point:
Hit the Delete key:
There are other ways...
- You could use the Scissors Tool or Knife tool to cut the path,then delete the end section
- You could draw a new line and use Pathfinder, then delete the end section
- You can select the path then use the Eraser Tool to erase art of it.
All the methods essentially involve the same amount of clicks/keypresses so no one is better than another.
But with any of them, you must have the path selected.
Based upon Question update. An effect has been applied. Therefore, you must expand the effect...
Object > Expand Appearance... or guess at where you want to cut the line and cut the actual path (the blue line), not the preview with the effect.