What do you mean by a distribution is homoscedastic (i.e. $ σ(Y|X = x) = σ$) in the context of simple linear regression?
Why do we need this assumption in simple linear regression?
What will happen to the regression if a distribution is not homoscedastic?
When you perform a regression, you are making assumptions about the distributions of the random variables whose outcome you have observed. Those observations are your data.
Homoscedasticity means that the distribution you assume is generating the $Y$ value of your data points has the same variance no matter the value of $X$.
Why do we need this assumption in simple linear regression?
The way you fit a simple linear regression model is that your look for the parameters that make the data you observed as likely as possible. This is called maximum likelihood estimation. The common recipe for finding those parameters (via algebra) works under the assumption of homoscedasticity.
Consider the following example:
Three data points are given and simple linear regression yields the following regression line:
Now, what if I told you that when $X$ takes the value $2$ the distribution of $Y$ has a very very small variance, same for the value $3$, while it has substantial variance given that $X$ takes the value $1$? In this case, assuming that the regression line is true, getting the data you got would be very unlikely, because the black dots are quite far from the line.
A regression line like this:
would give you a much greater likelihood.
