Birds are pretty, and it's fascinating to see the variety of sizes and shapes they come in. They do interesting things, and make cheerful sounds. In a world that sometimes can be dreary, birds are a delight to behold.
A second good answer is that birds around our homes are in fact a part of nature. They are not like dogs and cats, horticultural plants in our lawns, or trees planted along our streets, all of which are usually taken care of by people. Birds are free to roam wherever they wish and to do what is natural for them. Among the birds we can see with our own eyes, right in our own backyards, examples of how wild animals deal with seasonal changes, how they raise families, how they interact with one another and their environment, how they handle mankind's disruptions, how their appearances and behaviors reflect the general laws of nature, and much, much more.
Of course, this can also be said of other wild animals occasionally coming into our yards. However, birds are much more complex than insects and other arthropods, and there are many more different kinds of birds than, say, mammals, reptiles, and amphibians.
Last but not least, another good answer is that there are wonderful field guides enabling us to identify whatever we see, and innumerable books and other sources informing us about every species. Both birds, and bird information, are accessible, and bird watching is something doable.