How can insects survive in a world full of pathogens that are able to defeat the innate immune system?
There is not really a definitive explanation for why, although it's important to note that many mammalian pathogens are not adapted to insects and vice-versa. Insects need to survive insect pathogens, and they have a number of defenses for this purpose. Animals and their pathogens co-evolve and exert selection pressures on each other.
Killing the host (or killing the host quickly) is often not in the pathogen's best interest. Pathogens evolve towards their own successful reproduction as much as animals do. Success may ultimately mean evolving towards a less pathogenic state that improves a pathogen's chances of being transmitted.
As animals develop new defenses, they gain an advantage (relative to others of their own species) in surviving various pathogens. Pathogens evolve to escape these defenses, and a new equilibrium is established. An animal lacking this defenses is likely to be more susceptible to dying of infection, as the new equilibrium is dependent on them having these protections.
Here is an interesting paper on this view: http://hedricklab.ucsd.edu/PDF/HEDRICKImmunity113.pdf
As mentioned by InactionPotential, organisms and their parasites are caught in an arms race. When an organism develops a new defense, the parasites with traits that allow them to survive those defenses excel and vice versa. Parasites must balance their survival and reproduction with that of their hosts or go extinct. Over time they may become commensal/symbiotic, transfer to another species, and/or eradicate its host species.
Invertebrates' immune systems (including insects) as a whole are quite complicated. Because of the diversity of these systems we may not have anywhere near a complete understanding of them for decades. Some species intentionally upregulate their immune systems in anticipation of events that may expose them to pathogens. Social insect species, like ants, have evolved specialized immunities that take advantage of individual sacrifice, situational recognition, and colony health. Fruit flies possess radically expressive immunoglobulin receptors (IgSF), similar in some ways to the vertebrate major histocompatibility complex.
This diversity is key: the world is full of pathogens capable of defeating certain innate immune systems to some degree. However, the variety of challenges presented by even two different species usually prevents a pathogen from killing all of either species. As a pathogen becomes adapted to one species, it may lose its fitness for the other. If a pathogen is limited to one species, it has to make do with what it has or it will disappear with its host.
References and further reading
In the innate immune reponse, pathogens are recognized by a fixed repetoire of cell-surface receptors and soluble effector molecules. These receptors have evolved to recognize pathogens over hundreds of millions of years and provide a formidable defense against a wide range of pathogens. Adaptive immunity has evolved only in vertebrates and complements the innate system with a very different strategy that allows for an almost infinite number of adaptations in the production of B and T cells. That lower life forms survive without this adaptation is probably just a testimony to the efficiency of the innate system. Much of this answer must be credited to Peter Parham's text,The Immune System.
Insects have generally shorter life spans than many vertebrates, and since they are smaller they also have a lesser amounted of invested energy in the generation of individual members of the species. A large number of animals have nervous systems and are capable of movement, and can thus actively avoid environments that might be harmful to them and have predators that are either orders of magnitude greater or smaller in size than the organism in question.
If they have variation among individuals for self recognition and complement, then a greater ability to reproduce more different individuals quickly with a lesser amount of food may help fill the relative deficit in disease resistance from not having adaptive immunity.
For vertebrates that is generally not an option due to the longer life cycles of each individual organism in comparison with either insects or bacteria or viruses.