why would the free rider problem prevent a private business from investing in the building of a city sidewalk?

Because most of the public goods are operated by the government. The private charity does not work because of the problem of the free rider. An expensive public good is provided by the government. The free rider is the person who receives benefit for a good without paying.

The correct answer to this open question is the following.

The free-rider problem prevents a private business from investing in the building of a city sidewalk in that the private company invests its money to make a profit. And if the company's financial projection indicates that making money is not feasible, the company won't participate in the project.

A free-rider is the person who benefits from service without paying for it. So if the city sidewalk would be used mostly by free-riders and the company does not see a profit in the investment, absolutely it is not going to participate.