I think a good toy example here could be a market on a coinflip. Let's say heads pays out $1 and tails pays out $0, and people can buy and sell contracts that resolve this way. Naively, people might think that this is a 50/50 outcome, so if anyone wanted to buy or sell this contract for some reason, they'd likely be able to find someone to trade with them around the price of 50 cents.
Now, say that you somehow knew that this coin wasn't a fair coin, and instead was weighted 55% to fall on heads, 45% tails. Then, you would be happy buying these contracts for 50 cents -- given the contract pays out $1 if the coin lands on heads, and you know there's a 55% chance of heads, the expected price for the contract is 55 cents, and you make 5 cents in expectancy.
So, if you had information about the "fair value" of this contract, telling you that the price should be 55 cents, you'd be incentivized to buy the contract at prices below 55 cents. If the contract was trading at any price other than 55 cents, then, from your perspective, the price would be incorrect. And if the price is incorrect, then you'd be able to make money trading: buying for prices below 55 cents and selling at prices above 55 cents. And finally, as a result of your trading, you'd provide one-sided demand to the market, pushing the price closer to the actual correct price.
From this simple mechanism, wherein everyone who has information is incentivized to make money on their information by trading, prices start getting pushed to accurately reflect the aggregate of the information that everyone possesses! So in markets, there is a profit incentive to provide information, and this makes prices more accurate.
Finally, I think a common misconception is that prediction markets are sometimes wrong, as events priced at low probabilities sometimes happen. For example, in the 2024 election, Trump was trading at probabilities below 50%, but he still won! However, this is conflating present information about the future with future results. Given the earlier example about the biased 55% coin -- before we flip the coin, the best thing we can possibly say about the future really just is that there's a 55% chance of heads and 45% chance of tails. If we then flip the coin and it lands on tails, that doesn't mean we were incorrect -- it was just the best statement about the future that we could have possibly made.
Prediction markets -- or really any price system -- aggregate the best available information about the future. If you can confidently state that they are wrong -- that their best available information about the future is inaccurate -- then you should be trading and making money.
Does that clear things up a bit for you? It's a longer response, but I think it might address some of the confusion that you (or anyone else) might have regarding what people actually mean when they say that these prices predict the future.