Cycling is an inherently risky activity in terms of suffering a fall from your bike. It doesn't mean that children shouldn't do it, but it is a good idea to minimise the risk. Just as you wouldn't, for example, let a child go rock scrambling wearing flip- flops but would insist on them wearing shoes with a good grip, a helmet is a non-intrusive and appropriate piece of safety equipment suitable for the sport.
I am an experienced cyclist and have fallen off my bike twice in the last year- the first time was on grass, when I tried to cycle up a steeper slope than the tyres I have on my bike were suitable for. I didn't hit my head, but landed only about 2 ft from a tree stump, which if I had managed to cycle just one turn of the pedals further I may well have hit my head on. Personally, even though I doubt I would have caused myself serious injury, I would have been glad to avoid getting as big a bruise on my head as I did on my hip.
My second accident was just this weekend and was caused by a sudden mechanical failure when my chain jammed as I went into the lowest gear. My bike is well maintained - I took it into a bike shop for a service at the beginning of the week - but the day before it had fallen over, causing a problem not noticeable to the naked eye.
I was only travelling at a slow speed and was lucky that I was able to throw my weight so I fell onto a verge and not the road, but again, just a small change in factors and I could have been relying on my helmet for protection.
I also saw a man fall from his bike whilst riding on the grass last summer. A combination of swerving to avoid a loose dog, hitting a bit of a dip, and only grabbing the front brake saw him take a spectacular dive over the handlebars, straight onto his head. Fortunately, he was only going very slowly and was just left with a bruise, a headache and feeling a bit dazed, all which would have been reduced or avoided if he had been wearing a helmet.