To download and/or share images on iOS devices, follow the steps below:
Click on the "Download" or "Download HD" button next to the Image.
In the iOS alert click "View" (instead of download). Your Image opens in a new browser tab.
Use the touch and hold gesture on the image to access the contextual menu and select "Add to photos".
Your image is now saved in the library roll of your Photos App. You use or share it like any other photo in your Photos App.
Alternatively, you can click Download in the iOS Alert and your image will be downloaded to your Files App. See the yellow path in the image below for the reference.