Probably you're trying to use start.me on a touch device. We disabled this feature on touch devices because it caused other problems.
You can still move bookmarks from one widget to another, though.
- Open Edit mode on both widgets (sending and receiving), by clicking the downward-facing arrow in the top right corner and selecting Edit.
- Now drag and drop works as you would expect.
- After moving the bookmark, click Save in the context menu, or the Done button at the bottom.