I'm wondering how the people who work on or manage Bitcoin Core get paid, e.g. Wladimir J. van der Laan.
Do they do the work just... as volunteers?
The majority of Bitcoin Core developers are volunteers or started as volunteers. They worked on Bitcoin Core in their free time and contributed without anyone paying them.
Later, when a developer becomes more prominent and devotes more time to the project, they may seek out someone (a person or company) to sponsor them to continue to work on the project. Alternatively a person or company may wish to sponsor or hire an active and prominent Bitcoin Core developer so that they may contribute to Bitcoin Core's development and help to keep the software maintained and growing. Some are paid to work on Bitcoin Core full time, others to work on it part time.
Currently many of the most active contributors are either sponsored or employed to work on Bitcoin Core specifically. However many others contribute as volunteers in their free time and are not paid to work on Bitcoin Core.