I'm not looking for a definite answer to this question, but I'm rather interested in the different facts, studies, and discussions that exist regarding this subject.
In the early modern era, Sweden was known to be a deeply Lutheran country, so Lutheran in fact, that it was sometimes thought of (especially by the Swedish aristocracy and clergy) to be the ultimate defender of the Lutheran faith. I'm under the impression that it was still very religious at the start of the 20th century, however only half a century later, religion seemed to play a far smaller role in public life, and today, Sweden is amongst the least religious societies in the world.
I think it's especially interesting if one compares Sweden to the USA which also is Western, went through similar societal changes during the 20th century, but remained very influenced by religion and still is today. An interesting difference is that USA was a secular society from the start, whereas Sweden didn't become secular until 2000 when the church officially was declared not to be a state institution. Did religion perhaps play different roles in public life in the two countries already at the start of the 20th century?
I covered the relative secularization of Europe and America in an older post. But basically, the short social science/comparativist answer (rather than the particularist answer that I'd also be interested in hearing about Sweden's unique path) is that for countries where's only one religion and it has become relatively unimportant for national identity while being close to the state and there's no real existential security problems for the people living there, religion tends to be less important.
Sweden is probably not a very religious country, but I would argue that due to the particulars of Lutheranism it might seem even less religious than it really is. This is also the case for the other Lutheran countries - including Denmark even though Denmark is not even formally secular.
In Lutheran thinking there is the doctrine of "The Two Kingdoms", which holds that the domain of faith and religion is separate from the domain of worldly power. The church as an entity have generally stayed out of politics, and there are no important religious or nominally religious parties in the Scandinavian countries. Individual members of the clergy can and do engage in politics, but generally keep their faith separate from their political arguments.
Lutheran Christianity is also pretty averse to outward display of faith and to religious symbolism and imagery. Initially it was strongly iconoclastic - churches were cleaned of their interiors, statues destroyed and frescoes were painted over with white. This stance was somewhat lessened over time, but you still see much less overt religious symbolism and imagery. People don't have painting of saints or Jesus hanging on the wall in shops. There are no roadside alters. People don't draw crosses in the air or says Hail Marys. Priests only wear their uniform when performing services, not when walking the street. In Lutheranism faith is considered a deeply personal relationship between a person and God, not a public affair. My point is: If a Lutheran is strongly religious you probably wouldn't know unless you asked. Even if asked they might play it down. Faith is considered a gift by the grace of God to a wholly undeserving individual. So if someone doesn't have faith, there is no point in trying to convince them otherwise. Due to the doctrine of "sole fide" you are saved by faith and faith alone, not by performing good deeds or acting in a certain way. Therefore proclaiming your religion does not really make you seem like a better or more moral person to other people, which means there is not a lot of incentive to make a big deal out of it.
So given the lack of outwards indications of religion, how do you determine how religious Swedes actually are? You could ask them. Polled if religion is important in their life, Swedes scores relatively low compared to most other countries. But Sweden have a quite high Human Development Index (wealth, level of education, life expectancy), and taking into account the general correlation that countries with higher HDI has less religious populations, Sweden is not an outlier. So it doesn't really require an individual explanation. (The US is somewhat of an outlier though, in that people polls as more religious than in other countries with similar HDI. Youdatsrecist excellent answer explores this phenomenon.)
Of course you could ask why Sweden have achieved such a high HDI, and you could ask the really interesting (if contentions) question why higher HDI correlates with less religious population. But I think that should be a separate discussion.