How do Bayesian methods do automatic feature selection?
Your interpretation of L1 regularization sounds right. Is it used to perform feature selection? Yes, in the broad sense that this 'encourages' coefficients in the linear model to be 0, and those features with 0 coefficients are not used and can be removed.

Of course, this assumption about the prior distribution of coefficients is just an assumption you're adding. You are asserting that you need a lot of evidence to believe the coefficients are nonzero. I'm not sure it's the L1 regularization that's doing the feature selection, but your assumption, implemented by L1 regularization, that's leading you to conclude that some features do not contribute to the model.