Here's a use case: generating a report detailing the number of times that each student downloaded a specific file.
(NB - the Site Stats tool will only begin recording data only after it has been enabled. Therefore you should turn this tool on well in advance of running your first report.
To get started, visit Site Stats and select the "Reports" tab. Click "Add" and then, under the "Report" tab, be sure to click "Show" and add a title/description to your report (this is important if you wish to save this results for future reference).
In the "What?" section, select "Resources" and then check "Limit to resources". Browse to the file you wish to track and check the box next to its title. In the "When?" and "Who?" sections, use the pull down menu for each to show "All". In this case, since we want to see how many times each person downloaded that specific file, select only "user" in the "Totals by" section. Finally, choose the "Table" presentation method and click "Generate Report".
You will be presented with a table that shows how many times the file was downloaded by course members.
Play around with the options a bit (after you've generated a report, click the "Back" button and change a variable or two to see what you get). There's a lot of useful information in the data if you ask the right questions of it.
UNC has some more documentation on this topic here: http://sakaitutorials.unc.edu/module_build.php?tag=statistics&page=Site%20Stats#export