As you can see on the picture fro Swewi each party have their own ballots. You pick the ballots for the party you want to vote for (I believe there are blanks as well as Sweden has quite a lot of write-in parties and candidates).
In the past, you could take the ballot on the voting location where people could more or less keep an eye on you and at some locations party member were even giving them to voters. You could always take the ballots from more parties to hide your selection, but it really still wasn't that good.
But I understand that this part has been improved, not having the ballots in public.
The reason for having the individual ballots and not a long list like most other countries is that the election laws are very liberal in Sweden. It's very easy to have your party in the election. The full ballot would be terribly long!
Just look at some of the parties up for election: Donald Duck's party, The least bad party, The poor man's party, Capitalism with a friendly face and Evil chicken party. Not to mention a handful of more or less extreme parties on both wings.