Naively, I would expect it to mean that if you take sufficiently many predictions (i.e. there's one made every day), and you group them by predicted chance (70%, 80%, etc. at e.g. 10% granularity), then in each bin, the proportion of correct predictions should match the bin's assigned chance (e.g. between 75% and 85% for the 80% bin). And so given enough predictions, your expected probability for a single prediction coming true should approach the predicted chance. With more predictions, you can make smaller bins (to within 1%, etc).

"Probability of appreciable rainfall" * "fraction of specified area which will receive it" is 0.7. Or, I guess more properly it should be an integral over possible rainfall patterns. But "70% of London will definitely see lots of rain, and 30% will see none" and "we have 70% credence that all of London will see lots of rain, and 30% credence that no rain will fall in London" would both be reported as 70% chance of rain in London. https://en.wikipedia.org/wiki/Probability_of_precipitation