You can see that in the formulas, but it does make sense.
We have more information near the sample mean, than at x values far from .
For example if x is the age of a person when they can perform some task, then x=0 doesn't make any sense.
Hence a regression line of y(x) really is usless at x=0, but we do want an intercept in our model.
That way we are not forced to go through the origin.
BUT analyzing y at x=0 is pointless, so having a larger variance at x at 0 than at makes sense.
Look at http://www.weibull.com/DOEWeb/confid...regression.htm
when we set the variance term is minimized.
Likewise in the predition intervals