Thank you for using the Apple Support Communities to post your question. I see that you're wondering where pictures go when you tap "Save Image" on your iPhone. I am happy to help!
When you save images on your phone, either from a message, email, or some other source, they all go to the "All Photos" album in your iPhone's Photos app. To get there, open the Photos app, then tap "Albums" in the lower right corner. You should see the "All Photos" album there on the top left.
For more information about managing the pictures on your phone, check out the following resources:
Photos for iOS. Bringing all your memories into focus.
Photos overview - iPhone User Guide
Let us know if you have further questions, and have a great day!