Aseprite has two ways of making lines: The Line Tool, and the Pencil Tool.
With the Line Tool, you click and drag to make your line. The line will be drawn from where you clicked to wherever you eventually release the button.
With the Pencil Tool, you first click and release to place the start of your line, and then if you hold Shift afterwards, you’ll draw a line connected to the start. It’s a little unintuitive, but it’s currently the only way to draw lines with rounded ends or with other custom shapes.
The Curve Tool is for drawing curved lines. You click+drag to place the initial line, and then you click to place the two curve control points to curve it (the line you’ll get is previewed as you move your cursor). If you only want one control point, then double-click for the first one, that’ll place both points in the same place.