Displaying of the Excerpt is handled in your theme by use of the_excerpt() function. This function shows either what is in the post_excerpt field in your post or it truncates some amount of text from your post_content. The only part of this you would handle with Pods is making sure that the hand-entered Excerpt field is available for your Pods.
You can access this under Advanced Options for your post type under Supports, just check the box beside Excerpt:
If you reference this field in your Pods Templates, using the magic tag {@post_excerpt} it will show the actual value in the hand-entered Post Excerpt, not the results of the_excerpt function.