I don't know why people use fonts served from Google on their websites. Just serve the fonts from the server the site is on. It's like having javascript libraries served by 3rd parties; it's less robust.
I think it allowed browsers to cache font files across websites. But that might not even work anymore as I understand many browser vendors are moving towards resource isolation.
There still is the fact that you're getting fonts that are automatically subset into partial font files per character set, so if your pages mostly only use one or two character sets (like Latin and possibly Latin-extended), the browser only needs to download the font files for those particular character sets – at the same time you still retain the flexibility of using the full range of characters supported by that font if the occasion demands it, though.
(With Latin plus Greek plus Cyrillic and possibly some OpenType features like proper small caps you can get into the hundres of k range, and support for East Asian languages easily gets you into the megabyte range.)
Plus in theory fonts optimised for the respective combination of browser and OS. The former probably isn't as critical any more, as almost everything should support WOFF2 (or at the very least WOFF) these days, as for the latter – I know OSs each have their own font rendering peculiarities, but no idea how much the difference might be in practice.
I guess one advantage could be to better use browser caches for these things? If you visit multiple websites that include the same fonts or javascript libraries, the browser can reuse the cache for them.
