Navigating with Mouse
You can select issues, scroll up and down, as you would do with any table. Clicking on a link would usually open the link, so if you'd like to just select an issue, click anywhere in the row except on the underlined links. The row is also selected when you click on the JIRA actions icon at the end of the row.
To show sub-issues or hide sub-issues of an issue, click the Expander button near the issue summary.
To expand or collapse the whole hierarchy, use Expand All or Collapse All buttons 
on the toolbar.
If there are many issues in the structure, not everything is loaded from the server. As you scroll down or expand sub-issue lists, the data is loaded on demand, which means there might be a delay before the grid is filled with the data for the displayed issues.
Navigating with Keyboard
You can use arrow keys to select next or previous issue in the list. Left and right arrows expand and collapse sub-issue list.
To expand all sub-issues, press the Plus keyboard button twice. To collapse all sub-issue lists, press twice the Minus keyboard button .
Using Ctrl+Arrow Key moves the selected issue up or down in the hierarchy or indents/out-dents it.
You can press Alt+Down to open JIRA actions menu for the selected issue.
There are a lot more keyboard shortcuts that let you work with the Structure almost without touching the mouse. Press Ctrl+? to see the shortcuts cheat sheet or click Info at the bottom of the structure widget.