Adding to your explanation, the connection with Common Lisp's condition handling (which is a sort of generalization of exception handling) is that the side effects are something that happens in addition (or instead) of the expression returning a value. But if we want to do stuff besides returning, operationally we might just pause the evaluation of the expression, do our stuff, and then either resume it, or abort it and do something else.

So to evaluate a expression print("a") we pause evaluation, do the printing, and then resume it by returning () (or whatever value that print returns). So a language that has algebraic effects (or, equivalently, Common Lisp condition system) will be able to implement expressions that print to the terminal.