Pen mode is useful if you want to draw or write in the remote compter. The main difference between pen mode and regular mode is that in pen mode dragging a single finger across the screen results in a mouse left click drag operation (in regular mode you have to do a one-and-half tap to start dragging).

The best way to see this in action is to opening up a drawing program like Microsoft Paint on your computer and then dragging your finger across the canvas.