I live in Melbourne, Australia. I have cards with the state library, Melbourne city libraries, yarra city council libraries, and a membership with a small private library also in the city.

Firstly, books are still superior to e-readers (which i have and use), and far better than trying to read on a screen. I'll always get the physical copy if i can, and there's still a large number of books which don't have electronic copies. And apart from sci-hub, how else am i supposed to access half of the materials that universities subscribe to if i'm not a member of the library?

Secondly, they provide a quiet space in the city and community where you can go and dedicate to reading, investigation and study without requiring private space (which not everyone has): an egalitarian space. Open to everyone, neither differentiated by money nor access to private space, which is a premium in cities. The libraries variously provide classes, community groups, meeting facilities, IT and technology, television/video facilities, and event spaces. Yarra libraries has some television screens set up with gaming consoles which I notice the local refugee and public-housing community love to use. There's cafes and eating spaces in at least 3 of the library branches I use. They also provide regularly rotating exhibitions. One provides regular performance music and events.

I value libraries because they maintain an ideal: that knowledge is free, valuable, for the whole community, and that this is important to society, above and separate from private business and the pursuit of money.