In an earlier post of mine [2] (which includes an older revision of the timeline linked in [1]), I speculate that it was Facebook Chat that killed the mid-2000s chat networks of old like AIM, Yahoo Messenger, and WLM, rather than Google Talk or any particular missteps of those incumbent chat networks. For example, I was surprised to learn that AIM was present in the iOS App Store at launch on 2008-07-11 -- of course, there were no push notifications at the time -- not until 2009-06.
Facebook Chat went live 2008-04-06. Google didn't view them as a threat in messaging but a threat in social networking; they thought they had messaging in the bag with their 2008-09-23 release of Android 1.0, and their 2008-11-11 update to Google Talk which brought voice and video calling, and their 2009-03-11 acquisition of GrandCentral, which was soon rebranded an invite-only Google Voice. But Facebook kept growing and growing and it had an integrated chat on a website where people went to spend their time, instead of Gmail, where they went to manage email.
To combat Facebook on social networking, Google launched Buzz with aggressive auto-opt-in on 2010-02-09. Unfortunately a few months later on 2010-05-20, Android finally got push notifications, which enabled third-party messaging apps to be viable on Android. This led to the rise of cross-platform messaging network apps like WhatsApp, whose success came back to hurt Google later.
Buzz fizzled and attracted controversy for its aggressive piggybacking on Gmail, so Google tried again with Google Plus on 2011-06-28. That was a better effort, and it included the features "+Messenger", a text chat, and the video chat "+Hangouts". By this point Facebook had more than 700 million active users, and on 2011-08-09 they introduced a standalone app for just Facebook Chat, called 'Messenger'.
On 2011-10-12, Apple released iOS 5, which came with iMessages. This significantly improved the iOS platform's story on messaging, putting further pressure on Google. Having put many of its eggs into the Google Plus basket, it needed a win, which it didn't get.
On 2013-05-15, Google liberated Hangouts from Google Plus so that it could address messaging and social networking separately. Hangouts was being repositioned as the Android-default messaging app. This helped onboard new users, but on 2014-02-19 Facebook announced it was acquiring WhatsApp. At this point Google lost, and it began to flail about, sometimes reviving long-forgotten apps like Google Voice, sometimes deprecating integrations it had encouraged previously, and then making two more messaging apps.
Sources (in detail at):