I know that there are a number of companies which provide connections
to the PSTN so that their VOIP customers can receive calls on an
assigned public telephone number. How do these companies do it? Do
they just set up a PABX/router in each area code they service, or do
they make an arrangement with the telephone company whereby the
incoming telephone calls are fed straight into the internet?

Roughly how much would it cost to set up a similar system capable of
dealing with around a hundred distinct phone numbers and about twenty
incoming or outgoing simultaneous conversations?

Those companies usually have alliance with a local carrier, and pass
your VoIP calls to those carriers through a gateway or softswitch.

Max