When I first started practicing yoga, I never considered that it would involve standing on my head. And to be fair, headstands didn't feature for the first 15 years of my yoga practice. As I've said already, it's not all about the asana.
In my early 30s I became aware of and started practicing Ashtanga Yoga and related approaches (including vinyasa flow, where poses are creatively linked together in a flowing sequence throughout the practice). This is when headstands made their first appearance in my life.
To begin with, I pretty much refused to practice headstands. I often practice yoga alone, and having been mentally and physically fragile throughout most of my adult life, I didn't want to risk doing myself an injury. But over time, as I got stronger and more robust in body, mind and emotions, with the help of some wonderful Ashtanga teachers (and my now-husband) I began to experiment. First tripod headstand, then less supported variations.
I still find them very challenging, and the strength, co-ordination, stamina, balance and body tension required put me to the test every time. So perhaps that's part of the attraction- a sense of accomplishment and satisfaction when I don't simply wobble and fall over (or even just fail to get up there at all!).
It's not uncommon to hear headstand (a.k.a. sirsasana, in Sanskrit) referred to as something like 'the king of poses'. It's often applauded for myriad benefits including strengthening the shoulders and the core muscles, encouraging blood flow to the brain and awakening the crown chakra, through which we can experience greater connectedness with the universe.
Yoga teachers will also speak of the benefits of the benefits of going upside down in any yoga pose in terms of challenging our worldview. If things are feeling a bit stale in life, turning everything upside down in a (relatively!) controlled way as part of yoga practice might just be the shake-up you need. Equally, if everything around you seems to be going to hell, the opportunity to turn yourself upside down with it might give some well needed perspective.
In an Ashtanga practice, headstand comes towards the end of the primary series. This means you've got to get through a lot of other yoga poses, linked by the breath, before attempting it.
Other approaches to yoga postures, however, suggest that once you've got the hang of it, you can even start your asana practice with sirsasana (as long as it's followed by an appropriate counter-posture, preferably mountain pose a.k.a. tadasana).
Personally, I now like to practice headstands either at the end of a yoga posture sequence or as a punctuation to other things that happen in life. I work from home a lot, and have found headstand a great counter-point to a stressful day or meeting, or a useful break in between calls during a busy day. I don't even need to roll the yoga mat out to headstand, I just use the sheep rug that I otherwise keep for Kundalini yoga.
You may like to try headstands as part of your yoga asana practice if you judge they are appropriate for your body (and believe me, they weren't appropriate for mine for a long time!). The physical, spiritual and energetic benefits are vouched for by many long-time yoga practitioners.
Headstands however are not an essential part of yoga practice, so don't be put off if you think they're not for you.