Why do externalities lead to a Pareto-inefficient outcome? So we know that under a Pareto-inefficient outcome, there is room for Pareto improvement, i.e. making someone better off without hurting anyone.

Assume that there is a drug company which is in competitive equilibrium: and $$N^S=N^L\\
|MRS|=|MP_n|=|MRT_{(c,l)}|$$
$\pi$ is maximised etc. So everyone should be happy right? Since we are in a competitive market,

But if this drug company makes some pollution which is a threat to the health of citizens, we call this case a negative externality.

By definition if such negative externalities exist, even though we are in a competitive equilibrium, we would be Pareto inefficient. 

But how come we are Pareto inefficient? If we reduce the pollution somehow and reduce the externality, or create a market for pollution and let the pollution trade the firm would definitely be losing money since it would pay for the pollution. So isn't it impossible to make citizens better without making the firm worse off?
If you allow side payments then the issue you identify goes away in a Coase sense.

The citizens being polluted could pay for production to be reduced by one unit.  This amount would have to be between the benefit of the reduced pollution and the loss to the pharmaceutical firm from the reduced production. But this last figure is small: at the competitive equilibrium the firm is matching its marginal revenue with its marginal cost so its profit from the final unit of production is very small, while presumably the pollution effect is larger (if not then the competitive equilibrium is already efficient). 

So the position with the side payment dominates the position without it in a Pareto sense, as both the citizens and the pharmaceutical firm are better off.  Possibly further side payments and production reductions would be even more efficient, at least until the marginal profit loss exceeds the value of reduced pollution; that point is then Pareto optimal and the starting point was not. 

You may have to extend this slightly as the reduction in production also affects the customers of the pharmaceutical firm, who see a reduction in consumer surplus from the lower production. But again, side payments from the citizens to the customers can move to a Pareto optimal position, until the marginal value of the reduced pollution equals the combined marginal losses of the pharmaceutical firm and its customers.

Advocates of the "polluter pays principle" would describe this as unfair, but Pareto optimality does not judge fairness.  Another practical issue would be creating a mechanism which avoids free-riding and leads to an efficient agreement, a frequent issue with Coasian bargaining.