¡Hola Scott_C y Alme!
Subject pronouns (like yo, tú, ella, etc.) are quite flexible in Spanish and, as Alme says, can go on either side of the verb. Or, of course, they can be dropped altogether - as your instincts are telling you to do, Scott_C. As a result, there are three ways of saying this sentence:
Ésa es la razón que no salgo yo a pescar.
Ésa es la razón que yo no salgo a pescar.
Ésa es la razón que no salgo a pescar.
All three versions would be grammatically correct!
As you both know, native Spanish speakers include subject pronouns in their sentences either for clarity (which isn't really relevant in this case, since salgo is clearly in the yo form) or for emphasis. In the context of the lesson, Carmen says this line after Mauricio mentions all of the problems that befell him and his friend on their fishing adventure - so she's including the yo to emphasize that this is why she doesn't go fishing (in contrast with Mauricio, who does).
Now, the final big question here of course is why Carmen placed the yo after the verb - why not leave it in front of the verb? The answer is that it all comes down to flow: when native speakers include subject pronouns in their sentences, they place them either before or after the verb depending on what feels the most natural within that sentence. For this particular sentence, the words simply run more smoothly if you place that emphasizing yo after the verb - it's less intrusive, and doesn't break the sentence up as much. To get a better feeling for how this sentence flows in the conversation, it might be helpful to listen to the audio for this sentence in context in the main audio file (Mauricio starts talking at 00:29, and Carmen delivers this sentence at 00:40).
As always, if you still have any questions, just let me know! :)
Saludos,
Liss