Some years ago, like by 1998 or 1999, one would look at the pixel size (termed dot pitch) when purchasing a monitor. As of today’s standards (and let alone those Retina displays) pixels then were quite big. For example monitors were of the VGA kind, sporting 640 pixels horizontally. The usual size was 14″. Now screens in notebooks are about 14″ too, but they show 1280 or 1600 pixels instead of VGA’s 640, in about the same size.
This means that nowadays pixels are much smaller. If 1280px are displayed in (about) the same with where we saw 640px means that size is now ¼ of the original (half the side length, ¼ the surface). At 640×480 the serif fonts looked ugly, because the monitor was unable to render those tiny tips with enough detail. Anti-aliasing came later and helped a lot but text still didn't look sharp. Thus, we all preferred sans serif fonts.
Microsoft released Verdana as a font for screens. It renders slightly bigger than the other fonts, and also has a higher x-size well matched by its width that makes it quite readable. So everybody switched to sans serif fonts, and pages set in the IE default Times New Roman looked outdated. On the other hand, many
@media print stylesheets still specified Times because printer resolutions (in DPI) were much higher, and still are.
IE specified the same default fonts as Netscape did, to avoid breaking page layouts. Mozilla did the same later on. And all other browsers. Nobody takes the risk of making the change, and it doesn’t matter because nowadays almost everybody sets the font family and size for their sites.
So, IMO, we get a default of 13px because of historical reasons.
Choosing Arial is smart move. Check this web fonts statistics: Arial is in almost any computer out there. It comes with Windows and I think that with Macs too. There are equivalent fonts for Linux. Arial is surefire for a company that wants to deliver their service with the least trouble for the user.
As of the testing of fonts and font sizes for the web, I learned a lot by reading the Human Factors International newsletter. They ran readability tests on different font sizes and line lengths. But, as Mikko correctly pointed, it also depends on the reading distance.