I'm not sure if this is a bug or oversight but the Kagi Images section in the app does not let you download images. This is a big downside as it requires you to go directly to your browser app to do it instead, defeating the purpose of the Kagi app entirely.
If it is implemented, I feel it would work just like a mobile browser. You tap and hold on an image then a window will popup with an option to save the image to your device. By default (at least on Android since that's the platform I use), it should save to the 'Download' folder. If that's too disorganized for your liking then maybe images can be saved to a new directory under '/Internal storage/Pictures/Kagi'.