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!

As you get farther from $\bar x,\bar y$ uncertainty increases. There's fewer and fewer observations when you reach out to distant regions of the domain of your function.
The main source of uncertainty is the one about the slope of the line. Take a look at the drawing here. With the given sample of observations you can say that the best fit line should be somewhere between these two grey lines. The uncertainty around $(\bar x,\bar y)$ is the smallest, but once you step away from where the observations are located, uncertainty increases.

Here's how we can intuitively "derive" the asymptotic confidence interval, i.e. where $x^*$ is very far away from your observations. The confidence given by model MSE will proportionally expand as $|x^*-\bar x|\to\infty$. Think of it as approximate equality of ratios $\frac{MSE}{\sqrt n\sigma_x}\approx \frac{CI(x^*)}{|x^*-\bar x|}$. That's the asymptotic of your formula:
$$\lim_{x^*\to\infty} MSE \sqrt{\frac{1}{n} + \frac{(x^*-\bar x)^2}{\sum_{i=1}^n (x_i-\bar x)^2}} =\frac{MSE}{\sqrt n\sigma_x}|x^*-\bar x|$$

