Invasive.org explains who should care about invasive species.  Hikers, Campers:

Invasive plants, such as kudzu, English ivy and Japanese stilt grass can grow over trails and make them harder to follow and navigate through. Natural beauty is reduced by invasives. … Many invasive plants make hiking, camping, and other outdoor activities unpleasant. Exotic stinging needle … can take over camping sites, making it hard to find a good spot to camp.

Gardeners:

Landscape plants that seed freely, like privet … must be weeded out before they take over and displace plants that were painstakingly planted in your garden.

Consider also this review of Lyanda Huapt’s Crow Planet:

[Crows] may be the dark shape of our future. … She admits that she cannot quite love them. … Haupt also appreciates the birds’ intensely social biology. She tells stories of crow “funerals”, when the normally raucous birds gather silently around the body of a dead family member. She emphasises their startling intelligence. …

The crow’s ability to adapt to man-made environments – in contrast to the struggles of more fragile species – has made it one of the planet’s most successful bird species. But this achievement is the source of Haupt’s ambivalence: it’s everyone’s loss, she reminds us, if we create an environment that accommodates only tough survivor species like the crow.

While nature shows and nature lovers often give lip service to the wonders of natural selection, in fact they mainly love the particular species alive now.  When nature adapts to recent changes, nature lovers mostly disapprove.  Most folks react similarly when economic competition creates winners and losers; they say they approve of the competition that led to our current wealth, but they disapprove when new winners, e.g., Walmart or Borders, displace old losers, e.g., small stores.

In contrast, I’ve decided I mostly love the competitive process that produced these amazing things we see today.   So I expect to mostly approve of the future changes competition will bring.  Our descendants may not be beautiful to our eyes, but I expect them to be tough, smart, and scrappy, like the crow.

Added 2Oct: Mark Davis:

Only a few per cent of introduced species are harmful. Most are relatively benign; some, such as the honeybee, can even have beneficial effects. … With the exception of insular environments such as islands and lakes, there are very few examples of extinctions being caused by non-native species.