I learned mostly by doing, through undergraduate research, and then in graduate school. This was far more effective than lectures or books, though those are very helpful for getting started. I had one class that covered an overview of lots of the technology, and was lucky enough to have access to preprints of Koller & Friedman's book as a reference to fill in any gaps. I also read Judea Pearl's book on Bayesian Networks fairly early on.

As far as everyday reasoning, it made me somewhat more skeptical of long chains of A --> B, !B therefore !A, type of thing. It's easy enough to model this type of logic as a special case of PGMs. And the causal stuff is extremely useful for making me skeptical of arguments of the sort "If we did X, then Y would happen," and also how and when correlation is causality. Don't have any pat examples though, it's just something that infuses my thinking, such as learning about biological evolution.

I don't know how he learned, but I studied it through the very demanding, and worth every second, course from Coursera:

https://www.coursera.org/course/pgm