Hey a lot of people are having this issue. I was too. I updated to iOS 9.0.2. I was able to open the app and see my podcasts but after 5-15 seconds the app would crash again. I was able to get it to stop crashing though after going through a few steps.
1) keep opening the app until it stops updating your podcast list.
2) once all of your podcasts have stopped updating, double click on the home button to bring up the app switcher (not sure if that's what it's called but I hope you know what I mean)
3) close the Podcast app by sliding it up in the app switcher.
4) open the Podcast app again and click on the first podcast you see. The quickly hit the power button to lock your phone while the podcast is still playing. When you do this the app will stay open and keep playing that podcast.
5) let the podcast that you started listening to play all the way through and then let it start the next podcast in your list automatically (if you have any in queue).
When I did the above, the next time I opened the Podcast app it didn't crash and it hasn't since. Hope this works for you.