Assigning a probability to a one-off event is enumerating all the ways it could happen, all the ways it could not happen and assigning a probability to each of those ways. Obviously there is a lot of guesswork; but if you need to make a decision based on the future that approach gives you a much better chance of making a good decision. In practice an event will be made up of components that are more predictable than the whole, and some real uncertainties. There is a lot to be gained by thinking hard about the situation, and assigning a probability will do that.
Silly example - how do I estimate my risk of falling climbing up a set of stairs that I've never climbed before?
* Baseline risk of tripping - I have a lifetime of data.
* Increase of risk being on a staircase - I have an area I want to put my foot on (a specific step) that is about 1/3 of the area that my foot usually falls in, so that increases the risk by an amount that can be reasonably estimated.
* I will watch my foot - maybe an order of magnitude improvement in precision.
This is enough to let me estimate the risk of carrying a bulky object (that obscures my view of my feet) up a staircase. I've isolated the uncertainty (how much does visual observation change the odds) from the certainties (areas, background rate).
Now I can take that to several experts who will identify new mechanisms and tighten up my estimations on how big a deal the components are. In this way - even though the final % I come to would still be a bit arbitrary - it is starting to become a summary of what a large number of people think about the inputs to the problem and their relative magnitudes. Being able to communicate all that thinking with a single number is a miracle in its own way.
The data might favour one candidate, but even assuming it is unbiased and representative, it is only a random sample, and there is a chance it could be randomly wrong.
But the betting odds depend on how much knowledge you have - for someone with perfect knowledge the odds may be 0 or 1, if you know nothing you might guess 50:50 for practical purposes.
Kidding aside, your observation is correct, you need to perform repeated predictions. 538 does just that. They keep predicting a whole lot of outcomes, so you can check their track record. They even have a challenge for the audience, where you can record your own predictions and compare them with their corresponding predictions (see for example [1] for NFL games). The scoring of the predictions uses the Brier score [2], which is just a version of R-squared for classification problems.
In the case of the November 2016 election, other prediction sites were giving the Trump team a 1-2% chance, 538 was giving them a 20% change. Considering that, 538 comes out as the clear winner.
Which brings us to what I consider to be the best measure of prediction accuracy (my own invention, it doesn't have a scientific name yet): you can compare two sequences of predictions (let's say yours agains 538) by simulating bets at some mid-odds level. For example if I say Trump's probability of winning 2020 is 45% and you say it's 35%, then I'm willing to give you 2-to-3 odds, while you're willing to take that. We then see who stays ahead most of the time in this betting simulation.
For what is worth, here's a recent survey of different scoring types for predictions: "Assessing the performance of prediction models: a framework for traditional and novel measures" [3]
[1] https://projects.fivethirtyeight.com/nfl-predictions-game/ [2] https://en.wikipedia.org/wiki/Brier_score [3] https://www.ncbi.nlm.nih.gov/pubmed/20010215
One common mistake I've seen is people mistaking that 20% as the expected vote ratio. It definitely wasn't; that was hidden behind an apparently too-well-hidden toggle (buttons above the graph), and was floating around 45% Trump and 48% Clinton, with overlapping error ranges.
So, for example, when 538 said that Trump had a 30% chance of winning, its hard to say how accurate that was given it was a one time event. But if you look over all of their election predictions, and look at those they predicted to win 30% of the time, about 30% of them should have been victories (with some margin for error, of course). If 50% were victories, or only 10%, then maybe that would indicate their models aren't doing so hot.