Is it possible to play Podcasts through the headphones connected to an Apple Watch?
Yes, it's possible to play music directly from your Apple Watch.  It's limited, as far as I can tell, to the Music app on the Watch.  From reading the Apple page, I can't tell if you can play podcasts directly to the Watch or not.  Source



You can enjoy music on your Apple Watch without your iPhone when you pair with Bluetooth headphones. Open the Music app, scroll up, then tap your Apple Watch to choose it as the source. Then scroll back down and tap any of these options:


Now Playing. Play or pause what's currently playing on your Apple Watch.
My Music. After you sync a playlist to your Apple Watch, you can browse your music by Albums, Artists, and Songs. 
Playlists. Use the steps below to sync a playlist to your Apple Watch.


There's more info on how to create a Playlist on the Apple page that I've linked-to above.  You might need to download any music stored in the cloud, such as music from Apple Music or iTunes Match, to your iPhone before you can sync it with your Apple Watch.

This iMore article goes into more detail about the (convoluted) way to add podcasts to your Watch.  There's more detail there than can be added to this answer, unfortunately.