At the standard levels present in swimming pools, chlorine does not present any danger to eyes, within the true meaning of 'eyes'. i.e. there's no risk of damage to the surface of the eyeball or to the tissues inside the eye.
However, some people find that they suffer irritation and swelling to the upper and lower eyelids through contact with chlorine. (This is similar to the irritation and 'puffiness' experienced by many hay fever sufferers). Swimming clubs tend to 'play safe' with their rules (so that they don't get blamed for any such problems) but, for most leisure swimmers, goggles are probably totally unnecessary.
If there was any serious risk to eyes from chlorine, there would be representatives of firms like Claims Direct waiting outside every swimming pool in the country to find potential clients for lawsuits. You won't find them there, simply because chlorine (as used in swimming pools) is totally safe.
Chris