The first one, "They're Made Out of Meat" is a short story by Terry Bisson dramatized for "radio" by The Truth podcast.
The second is Jessamyn West's "The Lesson," which author Sherman Alexie read for the New Yorker fiction podcast.
What struck me about both "They're Made Out of Meat" and "The Lesson" is they're good stories. Even if you don't feel the way I do about veganism, you'll enjoy the stories just as much. What made them special to me was the way I heard them as an animal-lover. As a person who has chosen not to take part in a system that treats animals as objects. Both stories speak to this choice, but in different, subtle ways, and without being didactic (in fact, Terry Bisson says plainly he's not a vegetarian).
What makes them good is that they re-frame, within the process of storytelling, our assumptions and expectations, challenging us to think differently about the subject than we might have before. This is true of all great fiction, and part of why I love it so much. Great fiction teaches us more truth than facts, statistics, polls, and sound bites ever can.
Happy weekend and happy listening!