As you might know, the British occupied Iceland during the Second World War in order to secure Allied shipping lanes. While they were here, they built the Reykjavík Airport, Nissen huts, and parts of the road system.
However, something many people might not know is that Icelanders drove on the left side even before the Brits came. Even back then, this was something of an exception. Iceland’s former colonizer, Denmark, for instance, also drove on the right. The arrival of the British just cemented the habit, and it wasn’t until 1968 that Icelanders made the switch (and they might also have been influenced by Sweden, who made the switch in 1967).
Iceland was expanding its road system significantly during this time (the ring road was “only” completed in 1974), and the thinking was that before Icelanders spend all of the time and money building up their road system to meet modern standards, then any changes should be made before, not after, the project.
So, in anticipation of the last push to build the ring road, Icelanders made the switch in 1968.
Although the switch did officially happen on the appointed day, Icelanders had plenty of time to prepare. For quite some time before the big change, public service announcements instructed Icelanders about the new driving patterns. Some were also anxious at driving on the right at first, so the roads were actually rather empty in the first days of the switch, allowing everyone a little more time to get adjusted.