As human goods are traded around the world, insects (or other animals) sometimes unwittingly travel along. They go about life in their new location and sometimes, finding that their new circumstances suit them, they flourish. Depending on what amount of resources they need, they may just fit in. Or, people may begin to notice disruptions in natural communities—perhaps whole forests of oaks being stripped of their leaves, or local ground-nesting birds dwindling. Once people trace the disturbance back to the newcomer, it is labeled a pest!
Is that species a pest back home? Perhaps, but not necessarily. It's possible that in its continent of origin, it plays a harmonious role in natural communities. Something there (possibly predators or climate) may keep the insect’s population in check. (That’s natural processes at work.) Or it may be that plants and animals in natural communities back home are the offspring of individuals that survived a much earlier onslaught because they possessed characteristics that made it possible for them to co-exist with the insect. (That's natural selection.)
So an insect (or other animal) that is a contributing member of today’s natural communities in its continent of origin could be a pest running wild in another continent where it is non-native.