Apple has its own Podcasts app that lets you download podcasts on your iPhone and listen to them on an Apple Watch running watchOS 5 or higher. If you have an LTE-enabled Apple Watch, you can listen to podcasts without hopping onto a nearby Wi-Fi network.
Open the Podcasts app on your watch to swipe through your library of subscribed podcasts or individual episodes. From the Podcast player, you can pause, resume, jump ahead 30 seconds, go back 15 seconds, alter the speed, and change the volume.
You can also play podcasts from your phone using your watch as the controller. If you're not crazy about Apple's Podcasts app, alternatives include Downcast, Overcast, and Pocket Casts.