Living things (plants, animals, fungi, etc) evolved in communities. Plants turn sunlight and CO2 into sugars and starches that form the base of the food chain for all the other living things around them. Very slowly, those species that were best adapted to fit into their niche in the community survived and thrived. A type of equilibrium exists that changes slowly as the community itself changes.
Invasive species are really bad because they can seriously disrupt the existing plant and animal communities, often changing the character of their adopted ecosystem. Native species suffer, diversity is lost and the now out of balance community can become subject to damage from soil erosion and other maladies associated with a sick environment.
When species are dropped into a community where they did not evolve, some die off immediately because they cannot survive. Other species fit in more or less nicely, eating and being eaten by others; finding and inhabiting that new niche.
Some species, when introduced, into a new community are able to exploit the new home in ways that their new neighbors cannot. One example well known in the upper Midwest is garlic mustard. This plant sprouts from the ground in late February or March, just as soon as the snow melts. It continues growing through the warm months and into early winter, only going dormant in the deepest of the cold months. It also produces a tremendous amount of viable seed. Finally, its roots put out a chemical that inhibits other seeds from sprouting. Most of the micro organisms, insects and animals that would eat it in its native Europe do not exist here, so it has few natural enemies.
Hope that was helpful. Let me know if you have specific questions or need additional information.