I ran into this expression when reading a book on fish behaviour:
Predation may be reduced in a breeding group because of a dilution effect, whereby the likelihood of any individual being the target of a predator decreases as group size increases. Breeding group members may detect predators earlier than solitary animals, thus increasing their probability of escape. They may also be more successful in deterring or confusing predators through cooperative efforts with other group members. In addition, the synchrony of reproductive activities in a group may temporarily swamp the appetite of predators, leading to lower predation per capita when offspring are vulnerable.
More trivially, I noticed a review about a French restaurant:
The seven course tasting menu at €57 is outstanding value for money. This is not as daunting as it sounds as the portions don't swamp the appetite.
I have therefore assumed that in this context swamp could mean satisfy the appetite or something alike.
Both use definition #4, a common meaning of swamp:
3. (of a boat) to (cause to) sink or be filled with water:
The little boat was in danger of swamping.
The next huge wave swamped the boat.
4. to overwhelm:
I was swamped with work.
Meaning #4 was derived from meaning #3, and generally means something unpleasant.
In your first example, there are too many off-spring for the predators to eat. The predator's appetite is overwhelmed.
In your second, they are assuring you that you will be able to eat all the courses without feeling overly full; your appetite won't be overwhelmed. I haven't heard this particular expression before, but my appetite has been 'swamped' at holiday meals.
I don't think the implication is that "there are too many offspring for the predators to eat".
When the prey is grouped closely, in very large numbers, and moving around, it's very difficult for the predator to single out one victim and home in on it, especially when they all look alike and are of about the same size. It's seems paradoxical, but if you think about it, predators tend to single out an easy victim from among a smaller group (an immature, old, or wounded animal, for example).
There are too many individuals for the predator to hunt successfully.
You may be right, but I can't connect the difficulty in catching the fish to swamping the appetite.
They make the point that a smaller percentage of fish are eaten -- "leading to lower predation per capita when offspring are vulnerable." They aren't saying that the absolute number is smaller, but that the great number of fish is protective for the survival of a large part of the group.