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: 
Since you're asking about user stories, not tasks: A user story usually consists of several tasks that can be tackled by different team members, sometimes even different crafts. That means that, while you can certainly assign team members to any of the story's *tasks* (though you don't have to do that, either), **the story is more likely to be owned by the team as a whole**.

There can be exceptions, of course. For example, you could have a user story entirely made up of tasks that can only be solved by a specialist in the team, so it might make sense to assign the story to this person to make that visible.
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.