The podcast page takes the functionality and style of your WP theme's Archive page. It will look like any other Archive page on your site (think a Tag or Category page).
The /podcast page will populate with any episodes that you have set to be included in your podcast feed. This is by default the Podcast post type but can also include any type of post that you select from the top area in the Podcast - Settings - General area of your WP dashboard.
In the image above we've chosen to include regular Blog Posts into the Podcast RSS feed and /podcast page. By default any Podcast posts are included in the /podcast page and RSS feed IF they have a valid media file in the Podcast Episode Details area. Any post without a media file associated with it will be ignored by both the /podcast page and the podcast RSS feed.
If you would like to make changes to your /podcast page you will need to create a Custom Page Template, and we have another help doc which walks through some more of those steps here: Creating Custom Page Templates