Click and drag from one anchor point to the next, making sure that each new line segment begins where the last one ended. The Line Tool will snap onto the anchors.
Once you're done creating what looks like a closed path, the lines are still separate objects. To combine them into one path, select them all and press Ctrl/Cmd+J (for Join) or via the menu select Object > Path > Join.
The Pen Tool can be used in a similar manner, but each segment will be a continuation rather than a separate object. The last (closing) click will be indicated by a tiny circle on the mouse pointer.
If you make an open path, Ai will still "fill" it with a fill color by assuming a straight path between the last open anchor points. Individual straight paths, however, such as those drawn by the line tool, have no "inside", so they won't show the fill color even if you have assigned one.
Allen