On what device are you doing this?
What OS and version is it running?
By mp3 player you mean Apple's Music app, or some other 3rd party mp3 app?
If you are downloading podcasts from the Apple Podcasts store, they will be stored in the Apple Podcasts App, you don't need to drag it anywhere else. Podcasts are played directly in the Podcasts app.
click here ➜ Download, save, remove, and share podcast episodes on iPhone - Apple Support
Otherwise, please provide as much detail as possible about your issue, including specific device and OS version, and the actual apps involved and where exactly you are downloading the podcasts from. Otherwise it's impossible to offer any advice.