Suppose you fit a relationship between two series and want to flag when the residual is unusually large. The natural approach is to compute the residual's standard deviation and flag anything beyond some multiple of it. That approach systematically fails at exactly the moments it is supposed to catch.
The reason is that the sample standard deviation has a breakdown point of zero: a single extreme observation can move it without limit. So one large divergence inflates the very scale it is then divided by, and the resulting standardised residual is smaller than it should be. The bigger the break, the more it inflates the yardstick, and the more it understates itself.
Using MAD times 1.4826 in place of the sample standard deviation makes the flag work as intended. The largest divergence is measured against a scale set by the ordinary observations, not by itself, so a genuine break registers as a genuine break.
The median deserves the same treatment for the same reason. A mean residual is dragged by outliers; a median is not. Centring on the median and scaling by MAD gives a fully robust standardisation, and the cost is a small loss of efficiency when the data really is clean and normal, which oil residuals generally are not.
There is a limit worth stating. Robustness buys resistance to a minority of contaminated observations. It does not help when the underlying relationship has changed, because then the majority of recent observations are from the new regime and no estimator can tell you that the old fit is stale. Detecting a regime change is a different problem from resisting an outlier, and conflating them is how a robust model stays confidently wrong for months.