Once part of an ecosystem is altered, as by a fire or by being converted to someone's backyard, of course it doesn't stop being part of nature. If a forest is completely burned to ashes, very soon weedy herbs "invade" the new open area, months and years later woody shrubs and small trees appear, then larger trees, and perhaps in a hundred years or so, unless something stops it or the soil has been damaged too badly, a new forest like the old one will appear. This step-by-step replacing of one community by another is called succession.
Every stage of succession has its own particular species of plants and animals. In a study done in Georgia in 1956 by David Johnston and E.P. Odum, for instance, it was found that during the first two years after agricultural fields were abandoned, Grasshopper Sparrows and Meadowlarks were the main birds. In fields that had been abandoned for 15 years, tall grass and woody shrubs dominated, and the main bird present was the Field Sparrow, along with six other species. Fields abandoned for 35 years were occupied by young pine forests in which the main bird was the Pine Warbler, with nine other species. Where old fields had been abandoned for 150 to 200 years, pines had been replaced by oaks and hickories, and the main birds were Red-eyed Vireos and Wood Thrushes, along with 17 other bird species.
The point being made is that, from nature's point of view, our backyards -- even if they are nothing but concrete-covered alleyways -- are no more than natural areas in the process of succession. Nature's goal is to convert our backyards to the most complex, stable ecological community possible, given the backyard's climate and geological setting. If the first settlers in your area found forest in the spot now occupied by your backyard, your backyard is constantly in the process of reverting to a forest. If your neighborhood used to be prairie, it's reverting to that. Mowing, weeding, and pouring concrete continually set the succession process back.
Therefore, when a sprig of crabgrass appears in the crack of a slab of concrete in an alleyway, it's part of "nature's first-aid squad" rushing in to add nutrients and humus to the soil, and make conditions better for the next community of plants and animals. If the alleyway were to remain untouched for many years, eventually weeds would expand the cracks, send runners all over the bare concrete, form a vegetative mat atop the concrete, and a variety of birds would come to eat the weed seeds, and carry some seeds to other concrete-covered areas needing to be reclaimed. The fruiting head of a dandelion shown at the right may be upsetting to people who try to keep their grassy lawns manicured, but from nature's perspective this dandelion is nothing less than an heroic front-line battle doctor who has parachuted into enemy territory to help in the process of converting the awful, sterile, artificial lawn into a diverse, stable and beautiful natural area!
On a philosophical level, then, wild plants and animals occupy our backyards because it's their job to set the stage for greater natural diversity there. From an ecological perspective, it's because food and shelter for animals are present -- there are niches to be occupied.
You might also enjoy surfing to the site called Landscaping to Attract Birds, to see how your backyard habitat can be made to provide more food and shelter -- more ecological niches -- than it currently does...