The justification goes back to the work of Pigou in the early 20th century. The essence is this: the burning of fossil fuels creates a negative externality: a cost that is incurred, but not by that consumer, but by the general population, and for a century or so.
This in essence creates a kind of "social subsidy": the price does not fully reflect the costs. This means that fossil fuels are over-consumed, relative to the economically efficient quantity.
As I wrote on that other answer: the "Economics Help" site explains the issue
The deadweight loss from fossil fuel consumption is the red striped triangle: this comes about because fossil fuel producers do not have to pay for their pollution, leading to a competitive advantage, and an artificially high quantity sold Q1 at price P1.
This inefficiency can be corrected (in long-run equilibrium) if the externality is priced back in - which is what the carbon price is intended to do.
Given the presence of fossil-fuel incumbents, their lobbying power, and their decades of direct and indirect subsidies, it is highly likely that a carbon price set at the marginal social cost of carbon would be an insufficient signal. A very helpful one, but insufficient on its own.