How to Continue a Line Feature in QGIS?

In the next release (8 july 2016) this wil be fixed. Have a look at http://changelog.linfiniti.com/en/qgis/version/2.16.0/

There you can read:

Feature: Extend linestring geometries with reshape tool

The reshape tool now allows you to extend linestrings, by starting a reshape at either a line's start or end point.

This feature was developed by Martin Dobias

Untill then you can do what Mapperz suggests

The Digitizing Tools plugin has to amend line function

https://github.com/bstroebl/DigitizingTools/wiki/Usage

Prolong line

applies to: line layer (multi or single part)

Continue editing existing online features. Just click any end point of an existing line and digitize new vertices as if you would digitize the new feature.

In QGIS 3.4 (and probably before that), you can use the ‘Vertex’ tool (which is next to ‘Add line’ on the toolbar, by default). Choose it, then hover the mouse over the ending or starting point of a line, and you'll see a plus symbol next to it—this allows you to continue the line. Similarly, it allows you to add a new point between other two by hovering the cursor there and clicking on the plus symbol that appears.