Question: Let's suppose the development team is planning the next Sprint. What does it technically mean that a User Story is added to the Sprint Backlog? Is it assigned to a developer or to the whole team? (If it's assigned to the whole development team, then how do we do this in Jira?)

Answer: 
A "story" does not necessarily correspond 1:1 with a development task. You need to determine *from* the story what needs to be done, and by whom, in order to complete it.
As Bogdan already mentioned: "the team owns all of the stories", & the team should also owns the way to get those stories done. I mean pre-assigning to individual member or pulling when sprint progress are both fine.

One practice in my team is we always have individual owner at story level even when multiple members involve. The owner will be in charge (in front of the team) to facilitate all sub-tasks & relevant members to get the whole story done.