How do Story Points get burnt down and reflected on the Burndown Chart
The Burndown Chart actually reflects the burning of Story Points whenever an issue is resolved. Refer to the screenshot below for the actual Burndown chart in JIRA Agile:
As you notice, the example burndown chart will gather all the Story Points at the starting of the sprint and this is shown on the top left corner of the chart which in this case is 73, as the issue got completed (resolved) the value of the remaining changed burning down the value of the Story Point on the completed issue. In basic, during a sprint it will reflect the total capacity of the team working on the sprint and see the progress according to the time. It can be configured to be using time estimates as burndown value. Refer to the official documentation below for more information.
Note
As you can see as well, if you add issues into a sprint after the sprint start, it will still be captured in Burndown as Scope Change. However this might not be the case for other reports (example: Sprint Report, Velocity Report and etc)