More complete explanation of how confint works

At the bottom of the page for the function confint, under "Tips", it says, "To calculate confidence bounds, confint uses R-1 (the inverse R factor from QR decomposition of the Jacobian), the degrees of freedom for error, and the root mean squared error." I am not a statistician, so I'm hoping someone can show this explicitly. I am doing a simple least squares fit of y = mx+b. If you could please show the calculation of the confidence interval for the slope, m, as explained above and as a Matlab calculation (not just math), I would really appreciate it. Thanks.

[Charlie North]

This might help: https://uk.mathworks.com/help/curvefit/confidence-and-prediction-bounds.html