Thinking, Fast and Slow is full of illustrative experiments and examples that you can even try on yourself!
While we find it difficult to incorporate statistical information into our predictions, we have no such qualms using causal info. In fact, we will jump at causal info, expecting causal links where there are none. Framing the same question using causal instead of statistical info results in our drawing wildly different probabilities. System 1 likes a coherent story, and as such incorporates causes easily.
Regression to the mean is one such phenomenon that we have trouble understanding because we attribute causes to explain outliers, rather than attributing outliers to luck or randomness. The author cites numerous everyday examples where people are completely oblivious to regression to the mean, including in golf (scores on day 2 of a tournament are usually closer to the average than they were on day 1), the height of children, and the sports illustrated jinx (whereby athletes perform poorly after they appear on the cover - but of course, this is due to the fact that an athlete that is featured must have excellent results, no doubt aided by luck, in order to be on the cover to start with).
As such, our intuition must be tamed if we are to be accurate in our predictions. Otherwise, we are likely to exaggerate predictions away from the mean using causal explanations, not properly taking regression into account. Kahneman discusses a quantitative method to do this, which we can easily invoke using System 2.