The residuals are the differences between observed values and their predicted values. For a linear relationship (even with noise), the residual values should be consistent across the range of data. Graphing the residuals helps us qualitatively determine how well the data follows a linear relationship.
Even with good correlations, datasets with inconsistent residuals will inconsistently predict values of y. Often other relationship models will do a better job.
Sometimes it is tempting to say that variation in the x-variable causes the variation in the y-variable. This is not necessarily the case. Sometimes there are lurking (hidden) variables that are the underlying cause.
Here's a data set with two variables: life expectancy and number of televisions per person. Does more TV increase life expectancy? What are the lurking variables?
Sometimes one outlier in a data is too influential. If the observation is questionable, it should be removed from the data.
The range of effective predictions do not extend the range of the observed data!