I have uploaded an image and transformed it via the Media-Library, but can't see how to save the transformation

When using the "Transform" page of an asset, in the media library, the transformed image is automatically created and stored in your Cloudinary account as soon as they are delivered, i.e. when the URL is clicked.

This URL that features below the preview is the one to be used for embedding the image in your application.

Some extra information:
Once an image is uploaded to Cloudinary it's stored on your account as a resource, assigned a unique public-ID, and can no longer be edited (unless deleted or overridden). You can, though, generate various transformed versions based on the original image using Cloudinary's dynamic URLs (URL based API).

For example (public-ID couple):
Original uploaded image - http://res.cloudinary.com/demo/image/upload/couple.jpgFace-based cropped version - 
http://res.cloudinary.com/demo/image/upload/w_100,ar_1,c_crop,g_face/r_max/couple.jpg

In order to further simplify the process, these transformation URLs can also be constructed via the Media-Library or the integration client-libraries, e.g. - Rails, PHP, Node.js, Java (additional information about the integrations can be found here).

Furthermore, if you wish to edit (apply a transformation to) the original image you can re-upload your image and apply an incoming-transformation to it, in order for the transformation to take effect before the image is stored.