Inspecting residuals is how you find out whether a model is wrong in a structured way. If they fan out as the input grows you have heteroskedasticity. If today's residual predicts tomorrow's you have autocorrelation and are leaving information on the table. If they cluster around events you are missing a variable.
In a pairs trade, you regress A on B, take the residual series as the spread, and trade its z-score. The whole strategy is the bet that the residual is stationary and will return toward zero.
Practical check: plot residuals against time and against fitted values before believing any regression output. Ten seconds of looking catches errors that summary statistics hide.
Related: ols-regression, heteroskedasticity, spread, cointegration