I'm not a bird-watcher myself, but I have bird-watchers in my family, so I had rich opportunity to observe their conversations on their hobby. Here's a collection of points that I think contribute to the appeal of bird-watching. Note that for reasons of homogenuity, I'll use websites and quotes mostly from the UK as an example in the following, but the arguments should hold for most other temperate climates just as well.
(1) Changing availability encourages emotional investment
One aspect that sets birds clearly apart from other types of animals is their strongly seasonal pattern. True, in temperate climates, there are hardly any insects during winter, but the mammals that you can observe in the UK are observable throughout the whole year. With the migratory behavior of birds, you have immense fluctuations across the seasons: for many species, you'll be able to observe them only at specific times of the year. This seems to lead to emotional investment: You can look forward to the time when species A should be observable for the first time, and cherish if it eventually has returned from its winter quarters. Also, bird-watchers seem to develop forms of FOMO if a rare species B has migrated to an area where you'd not expect it normally.
(2) There are so many to collect
The Royal Society for the Protection of Birds claims that there are 574 bird species that can be observed in the UK. In comparison, the Mammal Society lists only 101 mammal species in the British Isles, and there are apparently only 6 reptile species in Britain. This means that while you may quickly have observed all reptile species that exist in your area, you will never run out of targets if you are a bird watcher.
(3) It's very accessible
Some birds are so common that you will recognize them without much training. The entries in the list of 19 common British birds are distinct enough that anybody should be able to identify at least a few entries already. This means that you can start bird-watching right away just by going to the local park, even with young children.
(4) There is room for personal development
You may start watching birds straight away, and even children should be able to reliably recognize Great tits and Blue tits. Yet, you need to invest some time, and you will require some experience, in order to be able to recognize a Marsh tit if you see one. I've been told that keeping waders apart, e.g. a knot from a redshank is not trivial, and that it can be a satisfactory experience to realize how many different species you can already spot.
(5) It's a social activity
As an outsider, I was surprised to learn how much of a social activity bird-watching actually is. If you're at a observation post, it's not rare that you'll be addressed by complete strangers who want to show you that rare species that they've just discovered. Bird-watchers are also well-organized, be it in traditional societies (such as the RSPB) or on social media, including dedicated networks like BirdTrack where you can record and share your bird observations.
Granted, some of these points might in principle apply to the observation of other vertebrates as well, especially the last one: I'm sure that there are societies dedicated to the observation of reptiles or of mammals – but there are just not as many. Other points are fairly unique to observing birds, I think.