Confidence band for simple linear regression - why the curve? I am really struggling to understand why confidence bands for regression lines have a curve to them. A few example plots showing curved CIs, taken from this post: Shape of confidence interval for predicted values in linear regression

I have read a bit and understand that the standard error for the sampling distribution of $\hat y$ at a given point ($\hat y^*$) is as follows:
$$ MSE \sqrt{\frac{1}{n} + \frac{(x^*-\bar x)^2}{\sum_{i=1}^n (x_i-\bar x)^2}} $$
Because the above equation has $(x^* - \bar x)^2$ in the numerator, the further we are from $\bar x$, the bigger the standard error will be, hence the curve being 'thinnest' when $x = \bar x$, and fatter as $x$ increases/decreases.
But why does that make sense?
My thinking is this: if one of the assumptions for a linear regression is homogeneity of variance for $Y$ across values of $X$, and the residuals from our regression are also equally distributed across values of $X$,  why would the variance of our sampling distributions of $\hat y$ not be the same at each value of $X$?
Clearly I am missing something!

There is 2 uncertainties here. As you mentioned there is the uncertainty with the slope thus the spreading curve at ends, but there is also an uncertainty at the mean. Yes, the curve is thinnest at the mean but it is not zero. Thus the uncertainty of the slope passing through the mean's distribution causes the estimate to be non linear and generates the above examples.
