There are several ways to listen to podcasts on your Apple Watch, using Apple’s Podcasts app and third-party apps, without having your iPhone nearby.
The Apple Watch app on an iPhone can sync podcast episodes directly to the Apple Watch app.
On your iPhone, go to Settings > Podcasts and turn on Sync Podcasts.
Open the Apple Watch app on your iPhone. Go to the My Watch tab. Scroll down and select Podcasts.