Well, there's lycra and there's lycra. What I think they're actually asking is not "why do you have to wear bike shorts and a technical wicking shirt with pockets in a visible colour", but "why do I keep seeing these guys whose shoes match their shorts match their jersey match their glasses match their helmets match their bikes, and all of it in horrendous neon colours?"
I had been cycling for years in more or less suitable athletic gear before I got into road bike riding, training and racing, and I found the entire matching team kit thing rather offputting. People do wear martial arts "pyjamas", but not in public, and men who would never ever wear anything tightfitting or bright-coloured otherwise are suddenly out there wearing what looks like hot pink and orange skin suits with matching shoes, if your team has particularly bad taste. It is a group fashion thing, and can be a bit alienating. Now, of course, I think it looks pretty cool, but it took a while.
I find the best answer is to compare it to running, or here, to cross country skiing. You CAN run or ski in regular clothing, if you really want to, but almost everybody realizes that if you're running or skiing to get somewhere swiftly, training clothes are practical.
Winter riding is much less about badassery and much more about bundle-uppery. - malkin
1995 Kona Cinder Cone commuterFrankenbike/Selle Italia SLR Lady Gel Flow
2008 white Nakamura Summit Custom mtb/Terry Falcon X
2000 Schwinn Fastback Comp road bike/Specialized Jett