How do I listen to podcasts on my Apple Watch?
Apple doesn't yet support this out of the box, so you've essentially got two options:


Option 1 involves using iTunes on a Mac/PC to change the metadata of your podcasts and adding them to a playlist and syncing this across
Option 2 is somewhat easier but does mean having to use third party apps


Option 1 

This is the longer option, but does mean not having to pay for any third party apps.


On your Mac/PC, open iTunes 
Find the podcasts you'd like to listen to on your Apple Watch
Select the podcast and press commandI (Note: You can select multiple podcasts if you want)
The Podcast Info window will appear (Note: If you selected multiple podcasts at Step 3, you will first be asked if you're sure you want to edit info for multiple items)
Click on the Options tab
Change the media kind field at the top from Podcast to Music
Click OK.
Create a playlist specifically for your Apple Watch
Add your podcast(s) to this playlist 
On your iPhone open the Watch app
Make sure you have My Watch selected at bottom-left
Swipe up and tap on the Music option
Tap on the + Add Music option
Select the playlist you created at Steps 8-9


Now you have a native way to play podcasts on your Apple Watch. My belief is Apple will create a Podcasts app in a future version of watchOS, but until then this is the only Apple way to listen to podcasts.

NOTE: The above steps also work for Audio Books, but not those that have Digital Rights Management (DRM) protection (such as Audible books, etc).

Option 2

WatchPlayer transfers podcast episodes directly to the Apple Watch so they can be played without an iPhone. 

Overcast used to provide this functionality, but for some strange reason this feature was removed before the Apple Watch Series 3 went on sale. My hope is that they'll offer this functionality again soon, especially since the Apple Watch has really started to mature as a product and the LTE models don't need an iPhone around while you go for a run etc.

And of course, there'll be other apps as well that offer this functionality.

NOTE: I am not affiliated in any way with either WatchPlayer or Overcast.