Subscribed podcasts are downloaded automatically if (1) you are on wifi as most podcast streams are probably over the cell data size limit (2) allow podcast app to run in the background (config in Battery Saver).
If you have sufficient cell data plan, you can also tap on the episode directly to start streaming and playback instead of waiting for it to download.
If you don't allow it to run in the background, you need to open the app and open the podcast you subscribed. Then it will start automatic download.