I started cycling in a limited way about 4 years ago when I was 76 pounds overweight. Now I'm about 20 pounds overweight. I feel soooo much better. I can climb stairs. I can walk. I can do the grocery shopping without aching knees. I did not lose the weight by cycling alone. I did Curves for a while. I lifted weights for a while. I joined Weight Watchers and did what they told me. That made the biggest difference. I have not starved myself, and I eat lots of really good food, but it's nutritious food, not empty calories, and it's filling.
I'm 60 years old, and thinner than I've been in years. The laws of physics still apply.
As for what people say, some are just rude. Ignore them or tell them to get lost. However, I think it's possible that you are misinterpreting, "You ride a bike!" sometimes. When I ride my bike to work, which I've done occasionally since I was my heaviest, people are amazed. You'd think I arrived by dogsled or hot air balloon, and some of them express that they think I must be very brave and strong to be cycling when I have a perfectly good car in the garage. Some people don't get it at all, and others clearly think I've lost my last bit of common sense. These reactions haven't changed much since I've lost weight.