Least squares
In practical situations, we must estimate β0, β1 and σ from a data set that we believe satisfies the normal linear model.
The best estimates of β0 and β1 are the slope and intercept of the least squares line, b0 and b1
Since b0 and b1 are functions of a data set that we assume to be a random sample from the normal linear model, b0 and b1 are themselves random quantities and have distributions.
Simulated example
The diagram below represents a regression model with a grey band. A sample of 20 values has been generated from this model and the least squares line (shown in blue) has been fitted to the simulated data. The least squares line provides estimates of the slope and intercept but they are not exactly equal to the underlying model values.
A different sample would give 20 different points and a different least squares line, so the least squares slope and intercept are random.