It wasn't so much stripping as asserting the reality of the situation, but it was a downward spiral that started back in the 18th century. Gustav III instituted the constitution of 1772, which gave himself absolute power, stripping parliament of a lot of the power and liberties it had assumed upon the death of Charles XII. Gustaf III himself was assassinated as part of plot from the nobility, and loss of power effectively began with the coup d'état (again by the nobility) in 1809 when Gustav IV Adolf was replaced by childless Charles XIII.

Charles XIII assumed power in a far weaker position than his predecessor, with the 1809 constitution dividing legislative power between the monarchy and the nobility lead parliament (which was split between nobles, burghers, the priesthood and the peasants). While the following Bernadotte dynasty kept executive power until 1975 (and at times even attempted to assert it), losing full legislative power meant that it was a losing battle, especially against the increasing popularity of the socialistic and liberal movements of the 19th century and onwards. While the king was still the supreme commander of the Swedish armed forces, parliament set the budget and decided on military expenditures. While the king appointed ministers, they were legally responsible to the parliament. As time passed, the monarchy took more and more of a passive role in politics, until a commission was set forth in the 1950s to redraw the constitution. This resulted in the 1975 constitution, which stripped the king of all but ceremonial power.

So tl;dr: The aristocracy gradually removed power from the king, and social reform gradually removed power from the aristocracy, leading to a crowned republic.