How do people end up writing a chapter for a book? Do you need to be invited to do so by an editor?

Yes, usually these kind of projects stem from the participation in conferences/ seminars etc. although these papers might not necessarily be published as conference proceedings, but take the form of edited books that include ad-hoc/ commissioned essays as well.

Taking part in conferences, organize sessions, making your work known to the wider audience, and networking, are always the best ways to have a chance to be part of these publications.

Sometimes you can find call for publications on the web sites of research institutions.

Depending in which field you work, you can also be the editor of a volume: approach a publisher/ prepare a publication proposal and invite scholars in your field to be part of it.

Typically, yes. I submitted a paper to a workshop at a conference where the end goal of the workshop was to put together an edited book based on the submissions. After the workshop I was asked to extend my 4 page posiition paper into a book chapter which was again peer reviewed, accepted and published. The whole process took a long time though, the book was published about 2.5 years after the initial workshop paper accepted.

OK, cool. Cheers both.