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).
So, you're taking the frequentist approach, the probability is the fraction of the times the event happened as n goes to infinity? But tomorrow is unique. It will never repeat again -- n is always equal to 1.
And, as mentioned in another reply, calibration and probability are different things.
Alternatively, what single concept from statistics would most improve people's interpretations of popular news and daily life events?