The media player used by Seriously Simple Podcasting is the media player that is built into WordPress itself and, as such, the styling of this media player is handled by WordPress along with your theme. All Seriously Simple Podcasting does is add the player to your episode content, but it does not have any effect on the styling or display.
However, you can easily customise the audio player’s appearance using the Media Player Style Kit plugin. The plugin adds style controls for the audio player to your site’s Customiser, allowing you to easily make the player suit your theme more effectively. It will work on any WordPress site, but it is built with Seriously Simple Podcasting in mind, so will work seamlessly with your podcast.