I suppose the first interest in native plants simply was walking through the woods growing up and learning the names of trees. Kudzu grows in the south in a remarkably invasive way and that would be my first awareness of non-native species.
When i moved to California, i was greeted by the rich ecosystems here, and many unfamiliar plants. I walked and biked to work by a creek and for a time i was a stream keeper. The program was starting an invasive plant mapping program, and i learned about how some plants have severe impacts on the environment. Arundo, the giant cane, sucks much more water out of the ground than the native plants: removing it allows many more plants to survive. I also learned about the native California buckeye. That small tree or large shrub looses its leaves in the summer, an adaptation to the dry season. I read an article about how creeks begin to flow again for a short while after buckeyes loose their leaves, as the plants are no longer pumping water from the water table.
Native plants are more, then, than just the plants that happened to grow in a place before "the Columbian exchange" -- the global trade that began with the discovery of the Americas and a new European route to China. (I'm listening to 1493 which addresses ecological impacts of the global trade.) They have evolved to be particularly efficient in collaboration with the soil, climate, and fauna of an area.
Non-native plants might live gently in a region: as streamkeepers the non-native cattails were accepted because they weren't aggressive (out-competing other species) and provided resources that were used by native birds. Other plants are like extractive imperialists, dominating the landscape, pulling out the resources, and exploiting the natives until all the resources are gone.
Why fight it? Ecologists coined the term Homogenocene to refer to this period by pointing at how ecologies are becoming homogenized. (Geologists' choice of Anthropocene seems to be winning the popularity contest.) What's wrong with plants moving to where they can be to top species in the ecosystem? Isn't that the whole point of evolution?
Long term, sure, there's no problem: life continues apace, new species will evolve to fill the niches created by the victors, etc, etc. But I'm selfish. I'm concerned with my species. I believe humans will lose in the shorter term, our term, as invasive plants push out the natives. We don't just loose the chance to marvel at diversity, but we loose the knowledge embedded in ecosystems. I think we can fairly compare an ecosystem as a data structure, not quite a library, but a system that carries much understanding about how to deal with certain requirements. Arundo and California Buckeyes deal with requirements differently: adaptation took different routes. If humans lost the information of the variety of ways to solve a problem by investigating a "why" in an ecosystem, it's similar to loosing Alexandria. We might discover the same information, but... well, it's not efficient.
I think of the efficiency as a selling point to folks with MBA trained sensibilities about value. Aesthetically, diversity and abundance is so pleasing to me, i can't help but assume that everyone would prefer a native forest to a kudzu mound. I can't help but feel there's a moral responsibility to ecosystems just as there is to indigenous cultures. I don't think i've learned to effectively analyze and point out those values, though.