Hi James Everett Pope,

Not sure will my answer help you, but here are a bunch of nice articles regarding Regression Analysis & Statistics, which may make statistics intuitive:

https://statisticsbyjim.com/regression/

Next, my latest project https://www.researchgate.net/project/CARBONARA having for a goal to develop the software solution (CARBONARA - Computer Aided Research Boosts Optimization, Numerical And Regression Analysis) which which covers a few branches of mathematics that are essential for engineering education. It is still under development, buts also it is FREE to download (https://github.com/Vrda-GF/Carbonara), use, share...

Kind regards,

Anton

James -

Do you just mean simple linear regression (one predictor, with or without an intercept term) using a straight line, or do you also mean a curve (linear in coefficients), or multiple linear regression? 

Scatterplots are very helpful.  You could have the regression "line" included.  For multiple regression, it may or may not be helpful to graph one predictor at a time.  The dependent variable would be on the y-axis in each case.  You should also graph y on the y-axis and predicted y on the x-axis in a scatterplot. 

With a simple linear regression you might highlight an (x, y) data point, circling or using an arrow to show the corresponding point's x value on the x-axis, and do the same on the y-axis, and circle or otherwise highlight the point itself, and also highlight the (x, y*) point on the line, where y* is the predicted y value.  This would show the point contributing to the regression, but also the predicted y value that would have been used if that data had not been available.  Actually, that is not exactly correct.  If that (x,y) data point had not been available, then the coefficient(s) estimated would have been slightly different estimates, and the line would have not been exactly where it is.  But it is important to point out the difference between y and y*, as that difference is the estimated residual,  often called the "error," though that is a little misleading as we are talking about a random variable.  Because y is a random variable, we say we "predict" rather than estimate it.  (Note that prediction here generally does not mean forecasting.) 

Often the points have larger estimated residuals with larger predictions.  This is called heteroscedasticity.  It means that when creating the regression lines, using least squares regression, we weight the larger points less, as there is more variance associated with them.  This is called weighted least squares (WLS) regression. 

If your audience wants to know more about the estimated residuals, then a graphical residual analysis would help.  You could plot y* on the x-axis and estimated residuals on the y-axis. There is a lot of good information online about interpreting such a graph.  (For example, the Pennsylvania State University has online course information available to all which might help.) 

Note that predicted y in unweighted least squares regression  (OLS, ordinary least squares) is usually represented by a y with a "hat" (^) over it.  I use y* for predicted y with weighted least squares (WLS) regression, as a well-known econometrician, G.S. Maddala, used that.  Note that OLS is a special case of WLS with equal weights, so the regression weight for OLS would be any constant. 

You could highlight a couple of points on graphs and go through examples, showing how those data on the scatterplots correspond to data tables.  Perhaps a slide with a regression on a scatterplot with the points and lines shown, and a small table on the same slide with a couple of color coded points on the table corresponding to points on the graph, would help. 

You could also add prediction intervals to the regression scatterplots.   

Well, it isn't clear if some of the above is too elementary or some might be too complex for whatever you want to do, so I have given you a little on a few different things. 

Best wishes - Jim 

Some of the suggestions made by James R Knaub can be automated with the help of ggeffects (https://strengejacke.github.io/ggeffects/articles/plotmethod.html) library in R. In most cases this library can be used to create a plot directly from the model and raw data can be added to the plots using standard ggplot2 (https://ggplot2.tidyverse.org) functions.

Anton, James and Usman, thank you all for taking the time to offer me such rich and detailed feedback on visualizing linear regression results! Many thanks!