What does The benefits of a public good do not have to be paid for by "free riders." mean?

"by" is actually very important here, and corresponds to 「に」

be paid = 受身形
be paid for (by～～） = ～の支払いが（～～に)される
～～に払われる = be paid by ～～.

We paid $10 to the driver: 運転手に＄１０を払いました。
The ride was paid for by us : 私達に支払いはされた。 (受身形)
The driver was paid by us (forなし！) : 運転手は私達に払われました。

Be careful of the target of "for" here. "paid for" is a fixed expression. If you break it apart, you get a different meaning:
I paid the bill for you - for you = あなたのために
I will (pay for) your meal = 食事の支払いをする。 "pay for" ～～の支払いをする
少し違和感 : I will pay for your meal for you : あなたのために食事の支払い。。。 "for + for" = 違和感。

The situation that your professor describes above regarding public goods has other names as well--freeloading is one that comes to mind, as does "parasite", "moocher", and "sponge". In the US, we sometimes have a problem that is the opposite of free-riding--that of being forced to pay for services that one does not use. For example, tax dollars are used to subsidize public roads and oil companies, but if you don't drive a car, you don't get to directly benefit from those services. 面白い話ですね。