* Eating poisonous food: Some fish eat things that are toxic, either from their environment or from other animals.
* Being hooked: Fishing hooks can injure a fish's mouth and lead to infection or starvation.
* Choking: Large objects or even smaller things that get lodged in the gills can suffocate a fish.
But, it's not the mouth itself that kills them directly. The mouth is just the entry point for these problems.
So, it's a bit of a play on words. It's like saying "The car dies by its engine" - the engine is important, but the car dies because of something that happened *to* the engine.