It's pretty much equivalent to "I always eat breakfast", but I guess DL is just trying to teach the literal difference between the verbs "Mieć" and "Jeść".
In Polish "mieć" does not mean eat. "zawsze mam śniadanie"- is not about eating breakfast, but more I always take breakfast with me.
I should have said that "I always have breakfast" and "I always eat breakfast" are pretty much equivalent in_English. I didn't mean to imply equivalence in Polish; quite the opposite, in fact.
Apparently ś is softer than sz. But this English speaker can't really tell the difference. Yet. ;-)
I had an English teacher, I think he was American, who always said my friend's name "Kasia" as kasza- buckwheat. :)