Advisor
You can add columns to the Alerts page by clicking the gear. As @Mike Moniz mentioned, reports is the way to go to get the CSV or PDF. Message is usually a multiline value that is created from other tokens so i'm guessing that's why it can't be added as a column on the page. ##MESSAGE## is a valid token for integrations.
Agreed that there should just be an "export to csv" option right on the page.
You could write an external alerting script that logs all alerts to an html, json, xml, or csv and pop that into a dashboard using the HTML widget. I think ##MESSAGE## is also available for external alerting. Or like @Mike Moniz suggested, you can get at it via the api. Shouldn't be too difficult to do with javascript, so it could theoretically be built as an HTML widget within a dashboard.