speculate how less competition for food helps increase the reproductive success of the remaining insects after a pesticide is applied.

Of course, the spraying of pesticides will allow any insects that are not killed to have a huge advantage in the competition for food.  This will help to make these insects have much more reproductive success that they otherwise would have.  If these particular insects survive because they are more immune to the pesticide than other individuals, this will lead (eventually) to their species being much less susceptible to the pesticide.  This would be very bad.

When many insects are killed off, there will be more food for the others.  As the others have more food and easier access to it, they will be healthier.  This will make them more likely to successfully mate and lay fertile eggs.