Google Plus was formed around one observation: most of the people on the web don't have URLs.
When people don’t have URLs, it’s difficult to make assertions about them.
For example, to show you which restaurants people you trust* have recommended in an area you’re visiting, a recommendation system needs to have a latitude + longitude for the area, a URL for each restaurant (solved by Google Places) and a URL for each person (solved, ostensibly, by Google Plus).
People might be leaving reviews in TripAdvisor, or Yelp, and there’s no obvious way to tie all those people together into any kind of coherent social graph. Even with Gmail, there's no way to say that the person you email is the same person who's left a review, unless they have a URL (i.e. a Google Plus account) that connects the email address and the reviewer account together.
One URL to rule them all
Google Plus has an extremely clever way of linking together all those accounts, which involves starting with one trusted URL (Google Plus account), linking to another URL (GitHub, say), then linking back from that URL to your Google Plus account to prove that you own the GitHub account and can write to it. Now that both of those URLs are trusted, either of them can be used as the basis of a new trusted connection: linking from the trusted GitHub URL to a Flickr URL, and then from the Flickr URL to the trusted Google Plus URL (or any other trusted profile URL), is enough to prove that you also own the Flickr account and can write to it.
The problem is (and the question “why” is an interesting one), even after people had their Google Plus account, they didn’t use it to post reviews. It’s mystifying why most businesses don’t have thousands of reviews. The problem, I think, is what each URL represents. Each of my online profiles on different sites is literally a different “profile”, and I only choose to link some of them together. I don’t consider all of them to be equivalent. Someone’s TripAdvisor persona - the one they present to hotels and B&B’s - is not necessarily the one they’d use for LinkedIn, or for writing a peer review of an academic article.
When Google tried to connect YouTube accounts to Google Plus accounts, and failed, it was because people felt that those personas were distinct, and wanted the freedom to do certain things on YouTube without having it show up on their “personal record” in Google Plus.
This also perhaps explains why people are wary of using Google Plus authentication to sign in to an untrusted site - they’re not so much worried about Google knowing where their accounts are, but also that the untrusted site might create a public profile for them without asking, and link it to their Google Plus profile.
Anyway, Google Plus is going away as a social network, and maybe even as a public profile, but the data’s still going to be connected together behind the scenes - perhaps using fuzzier, less explicit connections as a basis for recommendations and decision-making.
* Note that this doesn’t necessarily mean “friends”, or even “people you know” - it’s quite common to trust the recommendations of a large group of strangers more than a small number of friends (particularly when it’s a recommendation of what that person likes rather than what they think you’d like - perhaps people consider themselves to be closer to “average” than each of their friends individually?)