They've taken a talking point that is
shared between liberals, libertarians, communists, anarchists and some socialists/social democrats and that directly affect peoples lives (a huge percentage of people share files etc. and don't see it as something wrong) at a time when big media are trying to tell people they're all criminals.
That's probably key to their success - they can take voters from across the spectrum who don't have a voice at all in large parties (the closest in Sweden perhaps being the socialist/communist Vensterpartiet - Left Party - which also wants to significantly limit copyright) but who have liberal tendencies.
This has been brewing for a while, and it'll likely spread outside of Sweden, limited by non-proportional voting in some countries. What is particular about Sweden is that Sweden had one of the most liberal policies on copyright in Europe to start with - non-commercial copying for "personal use" (including sharing with friends) was (is it still?) legal. This is a right that was ingrained. So when these rights are being taken away, it's natural you get a backlash.