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: 
When the team plans a sprint, they pull product backlog items or user stories from the top of the backlog and into their sprint. The stories they selected to do that sprint now form the "[sprint backlog](https://www.scrumguides.org/scrum-guide.html#artifacts-sprintbacklog)".

Now, **concerning assignment of these stories... it depends**.

As part of the planning, developers also can decide who does what. Jake could assign himself Story 1, Jane could assign herself Story 2, and so on. That's a visual indicator of who does what work, but it's not actually mandatory. **The team owns all of the stories, no matter the individual who works on them**.

For example, I worked in a team where we didn't assign the stories. Almost all of us were full stack developers so we just took stories and worked on them as we became available. We knew who was working on what because we were in the same office, we communicated when taking new stuff to work on, and we synchronized and planned our work during the daily. So stories were moving from TODO to IN PROGRESS without being assigned to anyone. Jira has no problem doing that. The story just shows up as UNASSIGNED in Jira.

So it depends on what you want to do. If you find the information useful to see, you can assign a story. If not, the team will still know how to organize themselves and do the work.
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.