Increased load on your web server, but other than that zero! (You might even be able to mitigate the load with caching). It progressively enhances existing links if JavaScript is enabled.

There are other libraries that do similar things, such as InstantClick: http://instantclick.io/

Assuming you use the script as suggested, letting a 3rd party site know your stats (and users) sounds like a non-trivial downside.

I'd surmise the author is benevolent, but if this were to be turned into a business, some kind of data play seems like the trivial next step.

Thanks. The only downside is that you can expect twice the number of requests to your pages (only the pages themselves, not their assets).