There can be many reasons. As of today, my reason is simply because it's necessary. But that used to be very different when I was younger. I grew up wetting the bed and didn't stop until my late teens. I only ever had a plastic mattress cover so I constantly woke up cold and wet. I never considered any other forms of protection, in fact I wasn't aware that any other existed.
In the mid-90's I was surprised and quite taken aback by the first Goodnites commercial. It was dark, black and white, and captured exactly how I felt about wetting the bed. It was also the first time that I'd ever heard of protection like diapers for older people. I wanted desperately to not wake up cold and wet anymore. However, I never got the chance to use anything until just after I stopped wetting.
When I got my hands on some GoodNites (and very soon after Depends and Attends) I was astounded by their simplicity and functionality. I was completely smitten with the feeling of both being dry and warm. I first wore them during the day out of curiosity and found them to be just as functional during the day. I had always had a "small bladder" and could struggle to hold it for very long during the day. I found that simply wearing protection helped me relax and I stopped worrying about any accidents or always finding a toilet.
So, in short, it was the simplicity and functionality. It was the ability for something so discreet and unobtrusive (compared to other medical devices or prosthetics) to provide so much protection as well as mental relief. Those are still very much part of the reason I wear in public, just as much as the actual physical need these days.