That quote (and many of the answers) are right on about the advantages of using AJAX (or similar technologies) to reduce data transfer requirements and improve the responsiveness of a web site.
Whether that's an advantage of "Web 2.0" is arguable because "Web 2.0" has no commonly-agreed or widespread definition. It certainly does not have any application or protocol specification, like, say, XML 1.0 or HTML 4.01.
If "Web 2.0" means "uses AJAX", then, yes, it is an advantage of "Web 2.0" - but I know of many sites which use AJAX and have never presented themselves as being "Web 2.0". There are also a few out there which call themselves "Web 2.0", but only use full-page loads and no AJAX.
If "Web 2.0" means "social networking", "has a brushed-metal look and rounded corners", "uses bright colors, simple shapes, and a reflection effect on the logo", or any of the many other definitions of "Web 2.0" that people have been seen to operate under, then AJAX and its benefits have nothing to do with it.
"Web 2.0" is a marketing buzzword which, like any good buzzword, means something different to each person who uses it. It is not an actual technology.