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I can provide you a very simple example. You have a blog, with articles. On a multi-thousand word article, you know that traffic that spent You get the point, I hope. Access logs are useless for this kind of basic information.
Another very good example - there was a blog from someone involved in the UK GDS initiative, that described the case of a woman sitting with a handheld console in the waiting room of a government office. Initially you think she's just playing, but peeking over her shoulder, you see she's filling an application for unemployment. The browser on the handheld is horribly outdated, but she probably doesn't have another choice. Therefore knowing your audience, and adapting to it is crucial. From them:
> people now access GOV.UK in many different ways - 16,500 visits came from games consoles in the last month (Xboxes/PlayStations/Nintendos) - including 65 sessions from a handheld Nintendo 3DS (this is in 2015 https://gds.blog.gov.uk/2015/11/20/2-billion-and-counting/)
It entirely depends on what your website is doing if you need analytics or not. But a lot of orgs and people actually do, and can and should make use of them.