Get civiMail report by API
I think you're looking for the ReportTemplate.getrows API call.  You can see examples of its use in the API explorer's built-in help (also available on Github).  More examples exist in the tests folder, since the most common use of this API is to create tests for reports.

UPDATE: As @demerit points out, this answer isn't really correct - but it CAN be :)  All(?) of the info on this mailing report is also available from the "Mailing Summary" report (for the stats) and the "Mailing Detail" report (if you need drilled-down data on click-throughs etc.).  So I recommend Demerit's answer if you're writing an extension, but my answer if you need to collect this data over REST.