You must win the 'Longest Road' award 10 times in matches. This will most likely unlock whilst you collect VP for other achievements but it does not take much effort to unlock. The longest road award is given to the player with the longest single road on the map that is over 5 sections long. There are two ways to get this, either gather lots of wood and brick cards or hope you are lucky with development cards and get road building twice. The first method is probably easier and you will probably be doing this anyway to win matches.

When you start a match, position your first settlements in spots that will gain you wood and brick cards then wait until you have enough resources for a road and add it on to one of your settlements. Be sure that you have one long road and not a branching road or it will not count. Win ten matches with the award and you are done.

You can check how long your road is by pulling the right trigger, the number next to the road icon is the number of road sections you have in your longest chain.

~Int