To export all attachmets from a trello card:

Open the Card and Click "Share and More" in the bottom right corner
Click "Export JSON" in the resulting menu
Copy all text that appears on the screen and paste in the box below

This also works for an entire board, just click the "Show Menu" on top right of board.
Then click "More", "Print and Export" and "Export JSON" and copy the resulting text into the box above.