So what is Melodifestivalen (or "Mello" as it is also affectionately called) exactly? It is just like the other national Eurovision selection competitions, for example Festival di Sanremo in Italy or Uuden Musiikin Kilpailu in Finland. If one is unfamiliar with those, they all work more or less like Eurovision itself.
In other words, several singers perform an original song, they all get votes from both the jury and the audience, the scores are combined and one is crowned the winner. The winner then gets to perform in that year's Eurovision song contest.
Thus, betting on Melodifestivalen is in principle just like betting on any other competition: find out about the competitors and their odds, choose your candidates and how much you want to bet and then make your bets and hope for the best!
If you are not Swedish or in Sweden and betting on a Swedish competition seems far-fetched, there is no need to worry. First of all, the competition is fully accessible online everywhere in the world. Second, there are many websites that have the current odds for the competitors.
So even if one is unfamiliar with betting on Mello or other singing competitions, by following the bookmakers' predictions one can start to get a good idea of who to bet on.