Why is this the equation for the fitness of cooperators?
As Bryan Krause points out in the comments, looking at the units gives you the answer. The correct form of the equation is $f_{c}=\frac{b(i-1)}{N-1}-c$, where $f_{c}$ is the fitness of cooperators, $N$ is the total population size, $i$ cooperators, $b$ is the fitness benefit of cooperating and $c$ is the fitness cost of cooperating. It doesn't make sense to subtract $c$, which is a kind of relative cost in terms of offpsring production, from $N$, which is population size. It would be like subtracting the cost of a car from the speed of an aeroplane. It also makes sense from the viewpoint that we expect the the fitness of cooperators, $f_{c}$, to reduce as the cost of cooperating increases, but independently from the number of defectors in a population.
Also, in the paper, it gives the equation with square brackets:

then the fitness of cooperators and defectors, respectively,is given
by fC=[b(i−1)/(N−1)]−c